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Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsraval 18901 Lemma for srabase 18903 through sravsca 18907. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
 
Theoremsralem 18902 Lemma for srabase 18903 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝑁 < 5 ∨ 8 < 𝑁)       (𝜑 → (𝐸𝑊) = (𝐸𝐴))
 
Theoremsrabase 18903 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (Base‘𝑊) = (Base‘𝐴))
 
Theoremsraaddg 18904 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (+g𝑊) = (+g𝐴))
 
Theoremsramulr 18905 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = (.r𝐴))
 
Theoremsrasca 18906 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (𝑊s 𝑆) = (Scalar‘𝐴))
 
Theoremsravsca 18907 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = ( ·𝑠𝐴))
 
Theoremsraip 18908 The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = (·𝑖𝐴))
 
Theoremsratset 18909 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
 
Theoremsratopn 18910 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴))
 
Theoremsrads 18911 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (dist‘𝑊) = (dist‘𝐴))
 
Theoremsralmod 18912 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)       (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
 
Theoremsralmod0 18913 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑0 = (0g𝑊))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑0 = (0g𝐴))
 
Theoremissubrngd2 18914* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑1 = (1r𝐼))    &   (𝜑· = (.r𝐼))    &   (𝜑1𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 · 𝑦) ∈ 𝐷)    &   (𝜑𝐼 ∈ Ring)       (𝜑𝐷 ∈ (SubRing‘𝐼))
 
Theoremrlmfn 18915 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod Fn V
 
Theoremrlmval 18916 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
 
Theoremlidlval 18917 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊))
 
Theoremrspval 18918 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊))
 
Theoremrlmval2 18919 Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝑊𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
 
Theoremrlmbas 18920 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(Base‘𝑅) = (Base‘(ringLMod‘𝑅))
 
Theoremrlmplusg 18921 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(+g𝑅) = (+g‘(ringLMod‘𝑅))
 
Theoremrlm0 18922 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
(0g𝑅) = (0g‘(ringLMod‘𝑅))
 
Theoremrlmsub 18923 Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(-g𝑅) = (-g‘(ringLMod‘𝑅))
 
Theoremrlmmulr 18924 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(.r𝑅) = (.r‘(ringLMod‘𝑅))
 
Theoremrlmsca 18925 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅𝑋𝑅 = (Scalar‘(ringLMod‘𝑅)))
 
Theoremrlmsca2 18926 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))
 
Theoremrlmvsca 18927 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(.r𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅))
 
Theoremrlmtopn 18928 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))
 
Theoremrlmds 18929 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(dist‘𝑅) = (dist‘(ringLMod‘𝑅))
 
Theoremrlmlmod 18930 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
 
Theoremrlmlvec 18931 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ DivRing → (ringLMod‘𝑅) ∈ LVec)
 
Theoremrlmvneg 18932 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
(invg𝑅) = (invg‘(ringLMod‘𝑅))
 
Theoremrlmscaf 18933 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅))
 
Theoremixpsnbasval 18934* 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.)
((𝑅𝑉𝑋𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓𝑋) ∈ (Base‘𝑅))})
 
Theoremlidlss 18935 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)       (𝑈𝐼𝑈𝐵)
 
Theoremislidl 18936* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       (𝐼𝑈 ↔ (𝐼𝐵𝐼 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝐼𝑏𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼))
 
Theoremlidl0cl 18937 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → 0𝐼)
 
Theoremlidlacl 18938 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    + = (+g𝑅)       (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑋𝐼𝑌𝐼)) → (𝑋 + 𝑌) ∈ 𝐼)
 
Theoremlidlnegcl 18939 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋𝐼) → (𝑁𝑋) ∈ 𝐼)
 
Theoremlidlsubg 18940 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → 𝐼 ∈ (SubGrp‘𝑅))
 
Theoremlidlsubcl 18941 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝑈 = (LIdeal‘𝑅)    &    = (-g𝑅)       (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑋𝐼𝑌𝐼)) → (𝑋 𝑌) ∈ 𝐼)
 
Theoremlidlmcl 18942 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ Ring ∧ 𝐼𝑈) ∧ (𝑋𝐵𝑌𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
 
Theoremlidl1el 18943 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.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → ( 1𝐼𝐼 = 𝐵))
 
Theoremlidl0 18944 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
 
Theoremlidl1 18945 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑈)
 
Theoremlidlacs 18946 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)       (𝑊 ∈ Ring → 𝐼 ∈ (ACS‘𝐵))
 
Theoremrspcl 18947 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 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾𝐺) ∈ 𝑈)
 
Theoremrspssid 18948 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
 
Theoremrsp1 18949 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 1 }) = 𝐵)
 
Theoremrsp0 18950 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 })
 
Theoremrspssp 18951 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 = (RSpan‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐼) → (𝐾𝐺) ⊆ 𝐼)
 
Theoremmrcrsp 18952 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐹 = (mrCls‘𝑈)       (𝑅 ∈ Ring → 𝐾 = 𝐹)
 
Theoremlidlnz 18953* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ∃𝑥𝐼 𝑥0 )
 
Theoremdrngnidl 18954 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵})
 
Theoremlidlrsppropd 18955* The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))
 
10.8.2  Two-sided ideals and quotient rings
 
Syntaxc2idl 18956 Ring two-sided ideal function.
class 2Ideal
 
Definitiondf-2idl 18957 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 = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
 
Theorem2idlval 18958 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)       𝑇 = (𝐼𝐽)
 
Theorem2idlcpbl 18959 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → ((𝐴𝐸𝐶𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
Theoremqus1 18960 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r𝑈)))
 
Theoremqusring 18961 If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝑈 ∈ Ring)
 
Theoremqusrhm 18962* 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.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &   𝑋 = (Base‘𝑅)    &   𝐹 = (𝑥𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))
 
Theoremcrngridl 18963 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)       (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))
 
Theoremcrng2idl 18964 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)       (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
 
Theoremquscrng 18965 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑆𝐼) → 𝑈 ∈ CRing)
 
10.8.3  Principal ideal rings. Divisibility in the integers
 
Syntaxclpidl 18966 Ring left-principal-ideal function.
class LPIdeal
 
Syntaxclpir 18967 Class of left principal ideal rings.
class LPIR
 
Definitiondf-lpidl 18968* 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 = (𝑤 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
 
Definitiondf-lpir 18969 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 = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
 
Theoremlpival 18970* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
 
Theoremislpidl 18971* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
 
Theoremlpi0 18972 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑃)
 
Theoremlpi1 18973 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑃)
 
Theoremislpir 18974 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
 
Theoremlpiss 18975 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ Ring → 𝑃𝑈)
 
Theoremislpir2 18976 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈𝑃))
 
Theoremlpirring 18977 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ Ring)
 
Theoremdrnglpir 18978 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ LPIR)
 
Theoremrspsn 18979* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾‘{𝐺}) = {𝑥𝐺 𝑥})
 
Theoremlidldvgen 18980* 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.)
𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺𝐼 ∧ ∀𝑥𝐼 𝐺 𝑥)))
 
Theoremlpigen 18981* 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.)
𝑈 = (LIdeal‘𝑅)    &   𝑃 = (LPIdeal‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐼𝑃 ↔ ∃𝑥𝐼𝑦𝐼 𝑥 𝑦))
 
10.8.4  Nonzero rings and zero rings
 
Syntaxcnzr 18982 The class of nonzero rings.
class NzRing
 
Definitiondf-nzr 18983 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 = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
 
Theoremisnzr 18984 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
 
Theoremnzrnz 18985 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing → 10 )
 
Theoremnzrring 18986 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)
 
Theoremdrngnzr 18987 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
 
Theoremisnzr2 18988 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2𝑜𝐵))
 
Theoremisnzr2hash 18989 Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 18988. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵)))
 
Theoremopprnzr 18990 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
 
Theoremringelnzr 18991 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 = (0g𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)
 
Theoremnzrunit 18992 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ NzRing ∧ 𝐴𝑈) → 𝐴0 )
 
Theoremsubrgnzr 18993 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑆 = (𝑅s 𝐴)       ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing)
 
Theorem0ringnnzr 18994 A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
(𝑅 ∈ Ring → ((#‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))
 
Theorem0ring 18995 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.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 𝐵 = { 0 })
 
Theorem0ring01eq 18996 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.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 0 = 1 )
 
Theorem01eq0ring 18997 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.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
 
Theorem0ring01eqbi 18998 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.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐵 ≈ 1𝑜1 = 0 ))
 
Theoremrng1nnzr 18999 The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ NzRing)
 
Theoremring1zr 19000 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.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
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