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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsramulr 19001 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 19002 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 19003 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 19004 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 19005 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 19006 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 19007 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 19008 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 19009 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 19010* 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 19011 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod Fn V

Theoremrlmval 19012 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

Theoremlidlval 19013 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊))

Theoremrspval 19014 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊))

Theoremrlmval2 19015 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 19016 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(Base‘𝑅) = (Base‘(ringLMod‘𝑅))

Theoremrlmplusg 19017 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(+g𝑅) = (+g‘(ringLMod‘𝑅))

Theoremrlm0 19018 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 19019 Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(-g𝑅) = (-g‘(ringLMod‘𝑅))

Theoremrlmmulr 19020 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(.r𝑅) = (.r‘(ringLMod‘𝑅))

Theoremrlmsca 19021 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅𝑋𝑅 = (Scalar‘(ringLMod‘𝑅)))

Theoremrlmsca2 19022 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))

Theoremrlmvsca 19023 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(.r𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅))

Theoremrlmtopn 19024 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))

Theoremrlmds 19025 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(dist‘𝑅) = (dist‘(ringLMod‘𝑅))

Theoremrlmlmod 19026 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)

Theoremrlmlvec 19027 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ DivRing → (ringLMod‘𝑅) ∈ LVec)

Theoremrlmvneg 19028 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 19029 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅))

Theoremixpsnbasval 19030* 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 19031 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 19032* 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 19033 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → 0𝐼)

Theoremlidlacl 19034 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 19035 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋𝐼) → (𝑁𝑋) ∈ 𝐼)

Theoremlidlsubg 19036 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 19037 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 19038 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 19039 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 19040 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑈)

Theoremlidl1 19041 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑈)

Theoremlidlacs 19042 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 19043 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 19044 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 19045 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 19046 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 19047 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 19048 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 19049* 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 19050 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 19051* 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 19052 Ring two-sided ideal function.
class 2Ideal

Definitiondf-2idl 19053 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 19054 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 19055 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 19056 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 19057 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 19058* 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 19059 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 19060 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 19061 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 19062 Ring left-principal-ideal function.
class LPIdeal

Syntaxclpir 19063 Class of left principal ideal rings.
class LPIR

Definitiondf-lpidl 19064* 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 19065 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 19066* 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 19067* 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 19068 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑃)

Theoremlpi1 19069 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑃)

Theoremislpir 19070 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 19071 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 19072 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 19073 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ Ring)

Theoremdrnglpir 19074 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ LPIR)

Theoremrspsn 19075* 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 19076* 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 19077* 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 19078 The class of nonzero rings.
class NzRing

Definitiondf-nzr 19079 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 19080 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))

Theoremnzrnz 19081 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 19082 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Theoremdrngnzr 19083 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ NzRing)

Theoremisnzr2 19084 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 19085 Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 19084. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵)))

Theoremopprnzr 19086 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)

Theoremringelnzr 19087 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 19088 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 19089 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 19090 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 19091 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 19092 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 19093 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 19094 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 19095 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 19096 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⟩})))

Theoremrngen1zr 19097 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.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremringen1zr 19098 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.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)    &   𝑍 = (0g𝑅)       ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Theoremrng1nfld 19099 The zero ring is not a field. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ Field)

10.8.5  Left regular elements. More kinds of rings

Syntaxcrlreg 19100 Set of left-regular elements in a ring.
class RLReg

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