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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrspssp 21001 The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpanβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝐺 βŠ† 𝐼) β†’ (πΎβ€˜πΊ) βŠ† 𝐼)
 
Theoremmrcrsp 21002 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &   πΉ = (mrClsβ€˜π‘ˆ)    β‡’   (𝑅 ∈ Ring β†’ 𝐾 = 𝐹)
 
Theoremlidlnz 21003* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝐼 β‰  { 0 }) β†’ βˆƒπ‘₯ ∈ 𝐼 π‘₯ β‰  0 )
 
Theoremdrngnidl 21004 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ π‘ˆ = {{ 0 }, 𝐡})
 
Theoremlidlrsppropd 21005* The left ideals and ring span of a ring depend only on the ring components. Here π‘Š is expected to be either 𝐡 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 βŠ† π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) ∈ π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ ((LIdealβ€˜πΎ) = (LIdealβ€˜πΏ) ∧ (RSpanβ€˜πΎ) = (RSpanβ€˜πΏ)))
 
10.7.2  Two-sided ideals and quotient rings
 
Syntaxc2idl 21006 Ring two-sided ideal function.
class 2Ideal
 
Definitiondf-2idl 21007 Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
2Ideal = (π‘Ÿ ∈ V ↦ ((LIdealβ€˜π‘Ÿ) ∩ (LIdealβ€˜(opprβ€˜π‘Ÿ))))
 
Theorem2idlval 21008 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdealβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    &   π½ = (LIdealβ€˜π‘‚)    &   π‘‡ = (2Idealβ€˜π‘…)    β‡’   π‘‡ = (𝐼 ∩ 𝐽)
 
Theoremisridl 21009* A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
π‘ˆ = (LIdealβ€˜(opprβ€˜π‘…))    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝐼 ∈ π‘ˆ ↔ (𝐼 ∈ (SubGrpβ€˜π‘…) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐼 (𝑦 Β· π‘₯) ∈ 𝐼)))
 
Theoremdf2idl2 21010* Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
π‘ˆ = (2Idealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝐼 ∈ π‘ˆ ↔ (𝐼 ∈ (SubGrpβ€˜π‘…) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐼 ((π‘₯ Β· 𝑦) ∈ 𝐼 ∧ (𝑦 Β· π‘₯) ∈ 𝐼))))
 
Theoremridl0 21011 Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
π‘ˆ = (LIdealβ€˜(opprβ€˜π‘…))    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ { 0 } ∈ π‘ˆ)
 
Theoremridl1 21012 Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
π‘ˆ = (LIdealβ€˜(opprβ€˜π‘…))    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐡 ∈ π‘ˆ)
 
Theorem2idl0 21013 Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Idealβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ { 0 } ∈ 𝐼)
 
Theorem2idl1 21014 Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Idealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐡 ∈ 𝐼)
 
Theorem2idlelb 21015 Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21022. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
𝐼 = (LIdealβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    &   π½ = (LIdealβ€˜π‘‚)    &   π‘‡ = (2Idealβ€˜π‘…)    β‡’   (π‘ˆ ∈ 𝑇 ↔ (π‘ˆ ∈ 𝐼 ∧ π‘ˆ ∈ 𝐽))
 
Theorem2idllidld 21016 A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    β‡’   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘…))
 
Theorem2idlridld 21017 A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π‘‚ = (opprβ€˜π‘…)    β‡’   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘‚))
 
Theorem2idlss 21018 A two-sided ideal is a subset of the base set. Formerly part of proof for 2idlcpbl 21022. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
𝐡 = (Baseβ€˜π‘Š)    &   πΌ = (2Idealβ€˜π‘Š)    β‡’   (π‘ˆ ∈ 𝐼 β†’ π‘ˆ βŠ† 𝐡)
 
Theorem2idlbas 21019 The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   π΅ = (Baseβ€˜π½)    β‡’   (πœ‘ β†’ 𝐡 = 𝐼)
 
Theorem2idlelbas 21020 The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   π΅ = (Baseβ€˜π½)    β‡’   (πœ‘ β†’ (𝐡 ∈ (LIdealβ€˜π‘…) ∧ 𝐡 ∈ (LIdealβ€˜(opprβ€˜π‘…))))
 
Theorem2idlcpblrng 21021 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
𝑋 = (Baseβ€˜π‘…)    &   πΈ = (𝑅 ~QG 𝑆)    &   πΌ = (2Idealβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ ((𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷) β†’ (𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷)))
 
Theorem2idlcpbl 21022 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
𝑋 = (Baseβ€˜π‘…)    &   πΈ = (𝑅 ~QG 𝑆)    &   πΌ = (2Idealβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) β†’ ((𝐴𝐸𝐢 ∧ 𝐡𝐸𝐷) β†’ (𝐴 Β· 𝐡)𝐸(𝐢 Β· 𝐷)))
 
Theoremqus1 21023 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
π‘ˆ = (𝑅 /s (𝑅 ~QG 𝑆))    &   πΌ = (2Idealβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) β†’ (π‘ˆ ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1rβ€˜π‘ˆ)))
 
Theoremqusring 21024 If 𝑆 is a two-sided ideal in 𝑅, then π‘ˆ = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
π‘ˆ = (𝑅 /s (𝑅 ~QG 𝑆))    &   πΌ = (2Idealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) β†’ π‘ˆ ∈ Ring)
 
Theoremqusrhm 21025* If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
π‘ˆ = (𝑅 /s (𝑅 ~QG 𝑆))    &   πΌ = (2Idealβ€˜π‘…)    &   π‘‹ = (Baseβ€˜π‘…)    &   πΉ = (π‘₯ ∈ 𝑋 ↦ [π‘₯](𝑅 ~QG 𝑆))    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) β†’ 𝐹 ∈ (𝑅 RingHom π‘ˆ))
 
Theoremqusmul2 21026 Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘„)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ([𝑋](𝑅 ~QG 𝐼) Γ— [π‘Œ](𝑅 ~QG 𝐼)) = [(𝑋 Β· π‘Œ)](𝑅 ~QG 𝐼))
 
Theoremcrngridl 21027 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdealβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ CRing β†’ 𝐼 = (LIdealβ€˜π‘‚))
 
Theoremcrng2idl 21028 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdealβ€˜π‘…)    β‡’   (𝑅 ∈ CRing β†’ 𝐼 = (2Idealβ€˜π‘…))
 
Theoremqusmulrng 21029 Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21030. Similar to qusmul2 21026. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
∼ = (𝑅 ~QG 𝑆)    &   π» = (𝑅 /s ∼ )    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ™ = (.rβ€˜π»)    β‡’   (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Idealβ€˜π‘…) ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ([𝑋] ∼ βˆ™ [π‘Œ] ∼ ) = [(𝑋 Β· π‘Œ)] ∼ )
 
Theoremquscrng 21030 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
π‘ˆ = (𝑅 /s (𝑅 ~QG 𝑆))    &   πΌ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) β†’ π‘ˆ ∈ CRing)
 
10.7.3  Ideals of non-unital rings

Remark: Usually, (left) ideals are defined as a subset of a (unital or non-unital) ring that is a subgroup of the additive group of the ring that "absorbs multiplication from the left by elements of the ring", see Wikipedia https://en.wikipedia.org/wiki/Ideal_(ring_theory) (19.02.2025), or the definition 4 in [BourbakiAlg1] p. 103 and the definition in [Lang] p.86, although a ring is to be considered unital (and commutative!) here, see definition 1 in [BourbakiAlg1] p. 96 resp. the definition in [Lang] p. 83, or definition in [Roman] p. 20.

In contrast, the definition of (LIdealβ€˜π‘…), does not require the subset to be a subgroup of the additive group, as can be seen by islidl 20982. If 𝑅 is a unital ring, however, it can be proven that each ideal in (LIdealβ€˜π‘…) is a subgroup of the additive group of the ring, see lidlsubg 20988. This is not possible for arbitrary non-unital rings, because the proof uses the existence of the ring unity.

 
Theoremdflidl2rng 21031* Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrpβ€˜π‘…)) β†’ (𝐼 ∈ π‘ˆ ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐼 (π‘₯ Β· 𝑦) ∈ 𝐼))
 
Theoremisridlrng 21032* A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
π‘ˆ = (LIdealβ€˜(opprβ€˜π‘…))    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrpβ€˜π‘…)) β†’ (𝐼 ∈ π‘ˆ ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐼 (𝑦 Β· π‘₯) ∈ 𝐼))
 
Theoremrnglidl0 21033 Every non-unital ring contains a zero ideal. (Contributed by AV, 19-Feb-2025.)
π‘ˆ = (LIdealβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Rng β†’ { 0 } ∈ π‘ˆ)
 
Theoremrnglidl1 21034 The base set of every non-unital ring is an ideal. For unital rings, such ideals are called "unit ideals", see lidl1 20995. (Contributed by AV, 19-Feb-2025.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Rng β†’ 𝐡 ∈ π‘ˆ)
 
Theoremrnglidlmmgm 21035 The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ π‘ˆ is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdealβ€˜π‘…)    &   πΌ = (𝑅 β†Ύs π‘ˆ)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ π‘ˆ ∈ 𝐿 ∧ 0 ∈ π‘ˆ) β†’ (mulGrpβ€˜πΌ) ∈ Mgm)
 
Theoremrnglidlmsgrp 21036 The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ π‘ˆ is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdealβ€˜π‘…)    &   πΌ = (𝑅 β†Ύs π‘ˆ)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ π‘ˆ ∈ 𝐿 ∧ 0 ∈ π‘ˆ) β†’ (mulGrpβ€˜πΌ) ∈ Smgrp)
 
Theoremrnglidlrng 21037 A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption π‘ˆ ∈ (SubGrpβ€˜π‘…) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdealβ€˜π‘…)    &   πΌ = (𝑅 β†Ύs π‘ˆ)    β‡’   ((𝑅 ∈ Rng ∧ π‘ˆ ∈ 𝐿 ∧ π‘ˆ ∈ (SubGrpβ€˜π‘…)) β†’ 𝐼 ∈ Rng)
 
Theoremdf2idl2rng 21038* Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
π‘ˆ = (2Idealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrpβ€˜π‘…)) β†’ (𝐼 ∈ π‘ˆ ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐼 ((π‘₯ Β· 𝑦) ∈ 𝐼 ∧ (𝑦 Β· π‘₯) ∈ 𝐼)))
 
Theoremrng2idlsubrng 21039 A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ (𝑅 β†Ύs 𝐼) ∈ Rng)    β‡’   (πœ‘ β†’ 𝐼 ∈ (SubRngβ€˜π‘…))
 
Theoremrng2idlnsg 21040 A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ (𝑅 β†Ύs 𝐼) ∈ Rng)    β‡’   (πœ‘ β†’ 𝐼 ∈ (NrmSGrpβ€˜π‘…))
 
Theoremrng2idl0 21041 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ (𝑅 β†Ύs 𝐼) ∈ Rng)    β‡’   (πœ‘ β†’ (0gβ€˜π‘…) ∈ 𝐼)
 
Theoremrng2idlsubgsubrng 21042 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ 𝐼 ∈ (SubGrpβ€˜π‘…))    β‡’   (πœ‘ β†’ 𝐼 ∈ (SubRngβ€˜π‘…))
 
Theoremrng2idlsubgnsg 21043 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ 𝐼 ∈ (SubGrpβ€˜π‘…))    β‡’   (πœ‘ β†’ 𝐼 ∈ (NrmSGrpβ€˜π‘…))
 
Theoremrng2idlsubg0 21044 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   (πœ‘ β†’ 𝐼 ∈ (SubGrpβ€˜π‘…))    β‡’   (πœ‘ β†’ (0gβ€˜π‘…) ∈ 𝐼)
 
Theoremqus2idrng 21045 The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 21024 analog). (Contributed by AV, 23-Feb-2025.)
π‘ˆ = (𝑅 /s (𝑅 ~QG 𝑆))    &   πΌ = (2Idealβ€˜π‘…)    β‡’   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrpβ€˜π‘…)) β†’ π‘ˆ ∈ Rng)
 
10.7.3.1  Condition for a non-unital ring to be unital

In MathOverflow, the following theorem is claimed: "Theorem 1. Let A be a rng (= nonunital associative ring). Let J be a (two-sided) ideal of A. Assume that both rngs J and A/J are unital. Then, the rng A is also unital.", see https://mathoverflow.net/questions/487676 (/unitality-of-rngs-is-inherited-by-extensions).

This thread also contains some hints to literature: Clifford and Preston's book "The Algebraic Theory of Semigroups"(Chapter 5 on representation theory), and Okninski's book Semigroup Algebras, Corollary 8 in Chapter 4.

In the following, this theorem is proven formally, see rngringbdlem2 21067 (and variants rngringbd 21068 and ring2idlqusb 21070).

This theorem is not trivial, since it is possible for a subset of a ring, especially a subring of a non-unital ring or (left/two-sided) ideal, to be a unital ring with a different ring unity. See also the comment for df-subrg 20460. In the given case, however, the ring unity of the larger ring can be determined by the ring unity of the two-sided ideal and a representative of the ring unity of the corresponding quotient, see ring2idlqus1 21079. An example for such a construction is given in pzriprng1ALT 21266, for the case mentioned in the comment for df-subrg 20460.

 
Theoremrngqiprng1elbas 21046 The ring unity of a two-sided ideal of a non-unital ring belongs to the base set of the ring. (Contributed by AV, 15-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    β‡’   (πœ‘ β†’ 1 ∈ 𝐡)
 
Theoremrngqiprngghmlem1 21047 Lemma 1 for rngqiprngghm 21059. (Contributed by AV, 25-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    β‡’   ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ ( 1 Β· 𝐴) ∈ (Baseβ€˜π½))
 
Theoremrngqiprngghmlem2 21048 Lemma 2 for rngqiprngghm 21059. (Contributed by AV, 25-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝐡 ∧ 𝐢 ∈ 𝐡)) β†’ (( 1 Β· 𝐴)(+gβ€˜π½)( 1 Β· 𝐢)) ∈ (Baseβ€˜π½))
 
Theoremrngqiprngghmlem3 21049 Lemma 3 for rngqiprngghm 21059. (Contributed by AV, 25-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝐡 ∧ 𝐢 ∈ 𝐡)) β†’ ( 1 Β· (𝐴(+gβ€˜π‘…)𝐢)) = (( 1 Β· 𝐴)(+gβ€˜π½)( 1 Β· 𝐢)))
 
Theoremrngqiprngimfolem 21050 Lemma for rngqiprngimfo 21061. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    β‡’   ((πœ‘ ∧ 𝐴 ∈ 𝐼 ∧ 𝐢 ∈ 𝐡) β†’ ( 1 Β· ((𝐢(-gβ€˜π‘…)( 1 Β· 𝐢))(+gβ€˜π‘…)𝐴)) = 𝐴)
 
Theoremrngqiprnglinlem1 21051 Lemma 1 for rngqiprnglin 21062. (Contributed by AV, 28-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝐡 ∧ 𝐢 ∈ 𝐡)) β†’ (( 1 Β· 𝐴) Β· ( 1 Β· 𝐢)) = ( 1 Β· (𝐴 Β· 𝐢)))
 
Theoremrngqiprnglinlem2 21052 Lemma 2 for rngqiprnglin 21062. (Contributed by AV, 28-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝐡 ∧ 𝐢 ∈ 𝐡)) β†’ [(𝐴 Β· 𝐢)] ∼ = ([𝐴] ∼ (.rβ€˜π‘„)[𝐢] ∼ ))
 
Theoremrngqiprnglinlem3 21053 Lemma 3 for rngqiprnglin 21062. (Contributed by AV, 28-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝐡 ∧ 𝐢 ∈ 𝐡)) β†’ ([𝐴] ∼ (.rβ€˜π‘„)[𝐢] ∼ ) ∈ (Baseβ€˜π‘„))
 
Theoremrngqiprngimf1lem 21054 Lemma for rngqiprngimf1 21060. (Contributed by AV, 7-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    β‡’   ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ (([𝐴] ∼ = (0gβ€˜π‘„) ∧ ( 1 Β· 𝐴) = (0gβ€˜π½)) β†’ 𝐴 = (0gβ€˜π‘…)))
 
Theoremrngqipbas 21055 The base set of the product of the quotient with a two-sided ideal and the two-sided ideal is the cartesian product of the base set of the quotient and the base set of the two-sided ideal. (Contributed by AV, 21-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    β‡’   (πœ‘ β†’ (Baseβ€˜π‘ƒ) = (𝐢 Γ— 𝐼))
 
Theoremrngqiprng 21056 The product of the quotient with a two-sided ideal and the two-sided ideal is a non-unital ring. (Contributed by AV, 23-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    β‡’   (πœ‘ β†’ 𝑃 ∈ Rng)
 
Theoremrngqiprngimf 21057* 𝐹 is a function from (the base set of) a non-unital ring to the product of the (base set 𝐢 of the) quotient with a two-sided ideal and the (base set 𝐼 of the) two-sided ideal (because of 2idlbas 21019, (Baseβ€˜π½) = 𝐼!) (Contributed by AV, 21-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ 𝐹:𝐡⟢(𝐢 Γ— 𝐼))
 
Theoremrngqiprngimfv 21058* The value of the function 𝐹 at an element of (the base set of) a non-unital ring. (Contributed by AV, 24-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   ((πœ‘ ∧ 𝐴 ∈ 𝐡) β†’ (πΉβ€˜π΄) = ⟨[𝐴] ∼ , ( 1 Β· 𝐴)⟩)
 
Theoremrngqiprngghm 21059* 𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 GrpHom 𝑃))
 
Theoremrngqiprngimf1 21060* 𝐹 is a one-to-one function from (the base set of) a non-unital ring to the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 7-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ 𝐹:𝐡–1-1β†’(𝐢 Γ— 𝐼))
 
Theoremrngqiprngimfo 21061* 𝐹 is a function from (the base set of) a non-unital ring onto the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ 𝐹:𝐡–ontoβ†’(𝐢 Γ— 𝐼))
 
Theoremrngqiprnglin 21062* 𝐹 is linear with respect to the multiplication. (Contributed by AV, 28-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (πΉβ€˜(π‘Ž Β· 𝑏)) = ((πΉβ€˜π‘Ž)(.rβ€˜π‘ƒ)(πΉβ€˜π‘)))
 
Theoremrngqiprngho 21063* 𝐹 is a homomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RngHom 𝑃))
 
Theoremrngqiprngim 21064* 𝐹 is an isomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   πΆ = (Baseβ€˜π‘„)    &   π‘ƒ = (𝑄 Γ—s 𝐽)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RngIso 𝑃))
 
Theoremrng2idl1cntr 21065 The unity of a two-sided ideal of a non-unital ring is central, i.e., an element of the center of the multiplicative semigroup of the non-unital ring. This is part of the proof given in MathOverflow, which seems to be sufficient to show that 𝐹 given below (see rngqiprngimf 21057) is an isomorphism. In our proof, however we show that 𝐹 is linear regarding the multiplication (rngqiprnglin 21062) via rngqiprnglinlem1 21051 instead. (Contributed by AV, 13-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &    1 = (1rβ€˜π½)    &   π‘€ = (mulGrpβ€˜π‘…)    β‡’   (πœ‘ β†’ 1 ∈ (Cntrβ€˜π‘€))
 
Theoremrngringbdlem1 21066 In a unital ring, the quotient of the ring and a two-sided ideal is unital. (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π‘„ = (𝑅 /s (𝑅 ~QG 𝐼))    β‡’   ((πœ‘ ∧ 𝑅 ∈ Ring) β†’ 𝑄 ∈ Ring)
 
Theoremrngringbdlem2 21067 A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 14-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π‘„ = (𝑅 /s (𝑅 ~QG 𝐼))    β‡’   ((πœ‘ ∧ 𝑄 ∈ Ring) β†’ 𝑅 ∈ Ring)
 
Theoremrngringbd 21068 A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π‘„ = (𝑅 /s (𝑅 ~QG 𝐼))    β‡’   (πœ‘ β†’ (𝑅 ∈ Ring ↔ 𝑄 ∈ Ring))
 
Theoremring2idlqus 21069* For every unital ring there is a (two-sided) ideal of the ring (in fact the base set of the ring itself) which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 13-Feb-2025.)
(𝑅 ∈ Ring β†’ βˆƒπ‘– ∈ (2Idealβ€˜π‘…)((𝑅 β†Ύs 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring))
 
Theoremring2idlqusb 21070* A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
(𝑅 ∈ Rng β†’ (𝑅 ∈ Ring ↔ βˆƒπ‘– ∈ (2Idealβ€˜π‘…)((𝑅 β†Ύs 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring)))
 
Theoremrngqiprngfulem1 21071* Lemma 1 for rngqiprngfu 21077 (and lemma for rngqiprngu 21078). (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐡 (1rβ€˜π‘„) = [π‘₯] ∼ )
 
Theoremrngqiprngfulem2 21072 Lemma 2 for rngqiprngfu 21077 (and lemma for rngqiprngu 21078). (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    β‡’   (πœ‘ β†’ 𝐸 ∈ 𝐡)
 
Theoremrngqiprngfulem3 21073 Lemma 3 for rngqiprngfu 21077 (and lemma for rngqiprngu 21078). (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   π‘ˆ = ((𝐸 βˆ’ ( 1 Β· 𝐸)) + 1 )    β‡’   (πœ‘ β†’ π‘ˆ ∈ 𝐡)
 
Theoremrngqiprngfulem4 21074 Lemma 4 for rngqiprngfu 21077. (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   π‘ˆ = ((𝐸 βˆ’ ( 1 Β· 𝐸)) + 1 )    β‡’   (πœ‘ β†’ [π‘ˆ] ∼ = [𝐸] ∼ )
 
Theoremrngqiprngfulem5 21075 Lemma 5 for rngqiprngfu 21077. (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   π‘ˆ = ((𝐸 βˆ’ ( 1 Β· 𝐸)) + 1 )    β‡’   (πœ‘ β†’ ( 1 Β· π‘ˆ) = 1 )
 
Theoremrngqipring1 21076 The ring unity of the product of the quotient with a two-sided ideal and the two-sided ideal, which both are rings. (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   π‘ˆ = ((𝐸 βˆ’ ( 1 Β· 𝐸)) + 1 )    &   π‘ƒ = (𝑄 Γ—s 𝐽)    β‡’   (πœ‘ β†’ (1rβ€˜π‘ƒ) = ⟨[𝐸] ∼ , 1 ⟩)
 
Theoremrngqiprngfu 21077* The function value of 𝐹 at the constructed expected ring unity of 𝑅 is the ring unity of the product of the quotient with the two-sided ideal and the two-sided ideal. (Contributed by AV, 16-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   π‘ˆ = ((𝐸 βˆ’ ( 1 Β· 𝐸)) + 1 )    &   πΉ = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)    β‡’   (πœ‘ β†’ (πΉβ€˜π‘ˆ) = ⟨[𝐸] ∼ , 1 ⟩)
 
Theoremrngqiprngu 21078 If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
(πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))    &   π½ = (𝑅 β†Ύs 𝐼)    &   (πœ‘ β†’ 𝐽 ∈ Ring)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π½)    &    ∼ = (𝑅 ~QG 𝐼)    &   π‘„ = (𝑅 /s ∼ )    &   (πœ‘ β†’ 𝑄 ∈ Ring)    &   (πœ‘ β†’ 𝐸 ∈ (1rβ€˜π‘„))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   π‘ˆ = ((𝐸 βˆ’ ( 1 Β· 𝐸)) + 1 )    β‡’   (πœ‘ β†’ (1rβ€˜π‘…) = π‘ˆ)
 
Theoremring2idlqus1 21079 If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜(𝑅 β†Ύs 𝐼))    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   (((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Idealβ€˜π‘…)) ∧ ((𝑅 β†Ύs 𝐼) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝐼)) ∈ Ring) ∧ π‘ˆ ∈ (1rβ€˜(𝑅 /s (𝑅 ~QG 𝐼)))) β†’ (𝑅 ∈ Ring ∧ (1rβ€˜π‘…) = ((π‘ˆ βˆ’ ( 1 Β· π‘ˆ)) + 1 )))
 
10.7.4  Principal ideal rings. Divisibility in the integers
 
Syntaxclpidl 21080 Ring left-principal-ideal function.
class LPIdeal
 
Syntaxclpir 21081 Class of left principal ideal rings.
class LPIR
 
Definitiondf-lpidl 21082* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal = (𝑀 ∈ Ring ↦ βˆͺ 𝑔 ∈ (Baseβ€˜π‘€){((RSpanβ€˜π‘€)β€˜{𝑔})})
 
Definitiondf-lpir 21083 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR = {𝑀 ∈ Ring ∣ (LIdealβ€˜π‘€) = (LPIdealβ€˜π‘€)}
 
Theoremlpival 21084* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑃 = βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})})
 
Theoremislpidl 21085* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝐼 ∈ 𝑃 ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔})))
 
Theoremlpi0 21086 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ { 0 } ∈ 𝑃)
 
Theoremlpi1 21087 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐡 ∈ 𝑃)
 
Theoremislpir 21088 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ π‘ˆ = 𝑃))
 
Theoremlpiss 21089 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑃 βŠ† π‘ˆ)
 
Theoremislpir2 21090 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdealβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ π‘ˆ βŠ† 𝑃))
 
Theoremlpirring 21091 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR β†’ 𝑅 ∈ Ring)
 
Theoremdrnglpir 21092 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ LPIR)
 
Theoremrspsn 21093* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐡 = (Baseβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐡) β†’ (πΎβ€˜{𝐺}) = {π‘₯ ∣ 𝐺 βˆ₯ π‘₯})
 
Theoremlidldvgen 21094* An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝐺 ∈ 𝐡) β†’ (𝐼 = (πΎβ€˜{𝐺}) ↔ (𝐺 ∈ 𝐼 ∧ βˆ€π‘₯ ∈ 𝐼 𝐺 βˆ₯ π‘₯)))
 
Theoremlpigen 21095* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π‘ƒ = (LPIdealβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) β†’ (𝐼 ∈ 𝑃 ↔ βˆƒπ‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐼 π‘₯ βˆ₯ 𝑦))
 
10.7.5  Left regular elements. More kinds of rings
 
Syntaxcrlreg 21096 Set of left-regular elements in a ring.
class RLReg
 
Syntaxcdomn 21097 Class of (ring theoretic) domains.
class Domn
 
Syntaxcidom 21098 Class of integral domains.
class IDomn
 
Syntaxcpid 21099 Class of principal ideal domains.
class PID
 
Definitiondf-rlreg 21100* Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg = (π‘Ÿ ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘Ÿ) ∣ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘₯(.rβ€˜π‘Ÿ)𝑦) = (0gβ€˜π‘Ÿ) β†’ 𝑦 = (0gβ€˜π‘Ÿ))})
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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