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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrlmscaf 20601 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(+π‘“β€˜(mulGrpβ€˜π‘…)) = ( Β·sf β€˜(ringLModβ€˜π‘…))
 
Theoremixpsnbasval 20602* The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)
((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ π‘Š) β†’ Xπ‘₯ ∈ {𝑋} (Baseβ€˜(({𝑋} Γ— {(ringLModβ€˜π‘…)})β€˜π‘₯)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (π‘“β€˜π‘‹) ∈ (Baseβ€˜π‘…))})
 
Theoremlidlss 20603 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   πΌ = (LIdealβ€˜π‘Š)    β‡’   (π‘ˆ ∈ 𝐼 β†’ π‘ˆ βŠ† 𝐡)
 
Theoremislidl 20604* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝐼 ∈ π‘ˆ ↔ (𝐼 βŠ† 𝐡 ∧ 𝐼 β‰  βˆ… ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝐼 βˆ€π‘ ∈ 𝐼 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝐼))
 
Theoremlidl0cl 20605 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) β†’ 0 ∈ 𝐼)
 
Theoremlidlacl 20606 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) ∧ (𝑋 ∈ 𝐼 ∧ π‘Œ ∈ 𝐼)) β†’ (𝑋 + π‘Œ) ∈ 𝐼)
 
Theoremlidlnegcl 20607 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ 𝐼) β†’ (π‘β€˜π‘‹) ∈ 𝐼)
 
Theoremlidlsubg 20608 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) β†’ 𝐼 ∈ (SubGrpβ€˜π‘…))
 
Theoremlidlsubcl 20609 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
π‘ˆ = (LIdealβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) ∧ (𝑋 ∈ 𝐼 ∧ π‘Œ ∈ 𝐼)) β†’ (𝑋 βˆ’ π‘Œ) ∈ 𝐼)
 
Theoremlidlmcl 20610 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐼)) β†’ (𝑋 Β· π‘Œ) ∈ 𝐼)
 
Theoremlidl1el 20611 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) β†’ ( 1 ∈ 𝐼 ↔ 𝐼 = 𝐡))
 
Theoremlidl0 20612 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ { 0 } ∈ π‘ˆ)
 
Theoremlidl1 20613 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐡 ∈ π‘ˆ)
 
Theoremlidlacs 20614 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   πΌ = (LIdealβ€˜π‘Š)    β‡’   (π‘Š ∈ Ring β†’ 𝐼 ∈ (ACSβ€˜π΅))
 
Theoremrspcl 20615 The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐾 = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐺 βŠ† 𝐡) β†’ (πΎβ€˜πΊ) ∈ π‘ˆ)
 
Theoremrspssid 20616 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐺 βŠ† 𝐡) β†’ 𝐺 βŠ† (πΎβ€˜πΊ))
 
Theoremrsp1 20617 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (πΎβ€˜{ 1 }) = 𝐡)
 
Theoremrsp0 20618 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpanβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (πΎβ€˜{ 0 }) = { 0 })
 
Theoremrspssp 20619 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 20620 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 20621* 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 20622 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 20623* 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 20624 Ring two-sided ideal function.
class 2Ideal
 
Definitiondf-2idl 20625 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 20626 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)
 
Theorem2idlcpbl 20627 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
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 20628 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 20629 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 20630* 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))
 
Theoremcrngridl 20631 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 20632 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‘…))
 
Theoremquscrng 20633 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
π‘ˆ = (𝑅 /s (𝑅 ~QG 𝑆))    &   πΌ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) β†’ π‘ˆ ∈ CRing)
 
10.7.3  Principal ideal rings. Divisibility in the integers
 
Syntaxclpidl 20634 Ring left-principal-ideal function.
class LPIdeal
 
Syntaxclpir 20635 Class of left principal ideal rings.
class LPIR
 
Definitiondf-lpidl 20636* 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 20637 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 20638* 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 20639* 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 20640 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 20641 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 20642 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 20643 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 20644 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 20645 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR β†’ 𝑅 ∈ Ring)
 
Theoremdrnglpir 20646 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ LPIR)
 
Theoremrspsn 20647* 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 20648* 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 20649* 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))
 
10.7.4  Nonzero rings and zero rings
 
Syntaxcnzr 20650 The class of nonzero rings.
class NzRing
 
Definitiondf-nzr 20651 A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing = {π‘Ÿ ∈ Ring ∣ (1rβ€˜π‘Ÿ) β‰  (0gβ€˜π‘Ÿ)}
 
Theoremisnzr 20652 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 β‰  0 ))
 
Theoremnzrnz 20653 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ NzRing β†’ 1 β‰  0 )
 
Theoremnzrring 20654 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ NzRing β†’ 𝑅 ∈ Ring)
 
Theoremdrngnzr 20655 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ NzRing)
 
Theoremisnzr2 20656 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o β‰Ό 𝐡))
 
Theoremisnzr2hash 20657 Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20656. (Contributed by AV, 14-Apr-2019.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (β™―β€˜π΅)))
 
Theoremopprnzr 20658 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ NzRing β†’ 𝑂 ∈ NzRing)
 
Theoremringelnzr 20659 A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
0 = (0gβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐡 βˆ– { 0 })) β†’ 𝑅 ∈ NzRing)
 
Theoremnzrunit 20660 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
π‘ˆ = (Unitβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ NzRing ∧ 𝐴 ∈ π‘ˆ) β†’ 𝐴 β‰  0 )
 
Theoremsubrgnzr 20661 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑆 = (𝑅 β†Ύs 𝐴)    β‡’   ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ 𝑆 ∈ NzRing)
 
Theorem0ringnnzr 20662 A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
(𝑅 ∈ Ring β†’ ((β™―β€˜(Baseβ€˜π‘…)) = 1 ↔ Β¬ 𝑅 ∈ NzRing))
 
Theorem0ring 20663 If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (β™―β€˜π΅) = 1) β†’ 𝐡 = { 0 })
 
Theorem0ring01eq 20664 In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (β™―β€˜π΅) = 1) β†’ 0 = 1 )
 
Theorem01eq0ring 20665 If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 0 = 1 ) β†’ 𝐡 = { 0 })
 
Theorem0ring01eqbi 20666 In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝐡 β‰ˆ 1o ↔ 1 = 0 ))
 
Theoremrng1nnzr 20667 The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Baseβ€˜ndx), {𝑍}⟩, ⟨(+gβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩, ⟨(.rβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩}    β‡’   (𝑍 ∈ 𝑉 β†’ 𝑀 βˆ‰ NzRing)
 
Theoremring1zr 20668 The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) ∧ 𝑍 ∈ 𝐡) β†’ (𝐡 = {𝑍} ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremrngen1zr 20669 The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) ∧ 𝑍 ∈ 𝐡) β†’ (𝐡 β‰ˆ 1o ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremringen1zr 20670 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    &   π‘ = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) β†’ (𝐡 β‰ˆ 1o ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremrng1nfld 20671 The zero ring is not a field. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Baseβ€˜ndx), {𝑍}⟩, ⟨(+gβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩, ⟨(.rβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩}    β‡’   (𝑍 ∈ 𝑉 β†’ 𝑀 βˆ‰ Field)
 
10.7.5  Left regular elements. More kinds of rings
 
Syntaxcrlreg 20672 Set of left-regular elements in a ring.
class RLReg
 
Syntaxcdomn 20673 Class of (ring theoretic) domains.
class Domn
 
Syntaxcidom 20674 Class of integral domains.
class IDomn
 
Syntaxcpid 20675 Class of principal ideal domains.
class PID
 
Definitiondf-rlreg 20676* 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))})
 
Definitiondf-domn 20677* A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn = {π‘Ÿ ∈ NzRing ∣ [(Baseβ€˜π‘Ÿ) / 𝑏][(0gβ€˜π‘Ÿ) / 𝑧]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 ((π‘₯(.rβ€˜π‘Ÿ)𝑦) = 𝑧 β†’ (π‘₯ = 𝑧 ∨ 𝑦 = 𝑧))}
 
Definitiondf-idom 20678 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn = (CRing ∩ Domn)
 
Definitiondf-pid 20679 A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
PID = (IDomn ∩ LPIR)
 
Theoremrrgval 20680* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLRegβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   πΈ = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )}
 
Theoremisrrg 20681* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLRegβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐡 ∧ βˆ€π‘¦ ∈ 𝐡 ((𝑋 Β· 𝑦) = 0 β†’ 𝑦 = 0 )))
 
Theoremrrgeq0i 20682 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLRegβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑋 ∈ 𝐸 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) = 0 β†’ π‘Œ = 0 ))
 
Theoremrrgeq0 20683 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐸 = (RLRegβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) = 0 ↔ π‘Œ = 0 ))
 
Theoremrrgsupp 20684 Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.)
𝐸 = (RLRegβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ:𝐼⟢𝐡)    β‡’   (πœ‘ β†’ (((𝐼 Γ— {𝑋}) ∘f Β· π‘Œ) supp 0 ) = (π‘Œ supp 0 ))
 
Theoremrrgss 20685 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLRegβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   πΈ βŠ† 𝐡
 
Theoremunitrrg 20686 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLRegβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ π‘ˆ βŠ† 𝐸)
 
Theoremisdomn 20687* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 ))))
 
Theoremdomnnzr 20688 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn β†’ 𝑅 ∈ NzRing)
 
Theoremdomnring 20689 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn β†’ 𝑅 ∈ Ring)
 
Theoremdomneq0 20690 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) = 0 ↔ (𝑋 = 0 ∨ π‘Œ = 0 )))
 
Theoremdomnmuln0 20691 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  0 )) β†’ (𝑋 Β· π‘Œ) β‰  0 )
 
Theoremisdomn2 20692 A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &   πΈ = (RLRegβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐡 βˆ– { 0 }) βŠ† 𝐸))
 
Theoremdomnrrg 20693 In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &   πΈ = (RLRegβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ 𝑋 ∈ 𝐸)
 
Theoremopprdomn 20694 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ Domn β†’ 𝑂 ∈ Domn)
 
Theoremabvn0b 20695 Another characterization of domains, hinted at in abvtriv 20223: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsValβ€˜π‘…)    β‡’   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 β‰  βˆ…))
 
Theoremdrngdomn 20696 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ Domn)
 
Theoremisidom 20697 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
 
Theoremfldidom 20698 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.)
(𝑅 ∈ Field β†’ 𝑅 ∈ IDomn)
 
TheoremfldidomOLD 20699 Obsolete version of fldidom 20698 as of 11-Nov-2024. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅 ∈ Field β†’ 𝑅 ∈ IDomn)
 
Theoremfidomndrnglem 20700* Lemma for fidomndrng 20701. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Domn)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 βˆ– { 0 }))    &   πΉ = (π‘₯ ∈ 𝐡 ↦ (π‘₯ Β· 𝐴))    β‡’   (πœ‘ β†’ 𝐴 βˆ₯ 1 )
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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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