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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sramulr 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.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴)) | ||
Theorem | srasca 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.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) | ||
Theorem | sravsca 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.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) | ||
Theorem | sraip 19004 | The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = (·𝑖‘𝐴)) | ||
Theorem | sratset 19005 | Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴)) | ||
Theorem | sratopn 19006 | Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴)) | ||
Theorem | srads 19007 | Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴)) | ||
Theorem | sralmod 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) | ||
Theorem | sralmod0 19009 | The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 0 = (0g‘𝑊)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → 0 = (0g‘𝐴)) | ||
Theorem | issubrngd2 19010* | Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) & ⊢ (𝜑 → 0 = (0g‘𝐼)) & ⊢ (𝜑 → + = (+g‘𝐼)) & ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) & ⊢ (𝜑 → 0 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) & ⊢ (𝜑 → 1 = (1r‘𝐼)) & ⊢ (𝜑 → · = (.r‘𝐼)) & ⊢ (𝜑 → 1 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ∈ Ring) ⇒ ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) | ||
Theorem | rlmfn 19011 | ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
⊢ ringLMod Fn V | ||
Theorem | rlmval 19012 | Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) | ||
Theorem | lidlval 19013 | Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)) | ||
Theorem | rspval 19014 | Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊)) | ||
Theorem | rlmval2 19015 | Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet 〈(Scalar‘ndx), 𝑊〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | ||
Theorem | rlmbas 19016 | Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | ||
Theorem | rlmplusg 19017 | Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | ||
Theorem | rlm0 19018 | Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | ||
Theorem | rlmsub 19019 | Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ (-g‘𝑅) = (-g‘(ringLMod‘𝑅)) | ||
Theorem | rlmmulr 19020 | Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (.r‘𝑅) = (.r‘(ringLMod‘𝑅)) | ||
Theorem | rlmsca 19021 | Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
⊢ (𝑅 ∈ 𝑋 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | ||
Theorem | rlmsca2 19022 | Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | ||
Theorem | rlmvsca 19023 | Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | ||
Theorem | rlmtopn 19024 | Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅)) | ||
Theorem | rlmds 19025 | Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (dist‘𝑅) = (dist‘(ringLMod‘𝑅)) | ||
Theorem | rlmlmod 19026 | The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | ||
Theorem | rlmlvec 19027 | The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑅 ∈ DivRing → (ringLMod‘𝑅) ∈ LVec) | ||
Theorem | rlmvneg 19028 | Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.) |
⊢ (invg‘𝑅) = (invg‘(ringLMod‘𝑅)) | ||
Theorem | rlmscaf 19029 | Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | ||
Theorem | ixpsnbasval 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.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))}) | ||
Theorem | lidlss 19031 | An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐼 = (LIdeal‘𝑊) ⇒ ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵) | ||
Theorem | islidl 19032* | Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝐼 ∈ 𝑈 ↔ (𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼)) | ||
Theorem | lidl0cl 19033 | An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 0 ∈ 𝐼) | ||
Theorem | lidlacl 19034 | An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 + 𝑌) ∈ 𝐼) | ||
Theorem | lidlnegcl 19035 | An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝑁‘𝑋) ∈ 𝐼) | ||
Theorem | lidlsubg 19036 | An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (SubGrp‘𝑅)) | ||
Theorem | lidlsubcl 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) & ⊢ − = (-g‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (𝑋 − 𝑌) ∈ 𝐼) | ||
Theorem | lidlmcl 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) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼) | ||
Theorem | lidl1el 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 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ( 1 ∈ 𝐼 ↔ 𝐼 = 𝐵)) | ||
Theorem | lidl0 19040 | Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈) | ||
Theorem | lidl1 19041 | Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈) | ||
Theorem | lidlacs 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)) | ||
Theorem | rspcl 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 ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) ∈ 𝑈) | ||
Theorem | rspssid 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 ∧ 𝐺 ⊆ 𝐵) → 𝐺 ⊆ (𝐾‘𝐺)) | ||
Theorem | rsp1 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 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐾‘{ 1 }) = 𝐵) | ||
Theorem | rsp0 19046 | The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 }) | ||
Theorem | rspssp 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 ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ⊆ 𝐼) → (𝐾‘𝐺) ⊆ 𝐼) | ||
Theorem | mrcrsp 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) | ||
Theorem | lidlnz 19049* | A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { 0 }) → ∃𝑥 ∈ 𝐼 𝑥 ≠ 0 ) | ||
Theorem | drngnidl 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 = (0g‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵}) | ||
Theorem | lidlrsppropd 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.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ⊆ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) | ||
Syntax | c2idl 19052 | Ring two-sided ideal function. |
class 2Ideal | ||
Definition | df-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‘(oppr‘𝑟)))) | ||
Theorem | 2idlval 19054 | Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐽 = (LIdeal‘𝑂) & ⊢ 𝑇 = (2Ideal‘𝑅) ⇒ ⊢ 𝑇 = (𝐼 ∩ 𝐽) | ||
Theorem | 2idlcpbl 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) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))) | ||
Theorem | qus1 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 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈))) | ||
Theorem | qusring 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 ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring) | ||
Theorem | qusrhm 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 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~QG 𝑆)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈)) | ||
Theorem | crngridl 19059 | In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂)) | ||
Theorem | crng2idl 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)) | ||
Theorem | quscrng 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 ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing) | ||
Syntax | clpidl 19062 | Ring left-principal-ideal function. |
class LPIdeal | ||
Syntax | clpir 19063 | Class of left principal ideal rings. |
class LPIR | ||
Definition | df-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})}) | ||
Definition | df-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)} | ||
Theorem | lpival 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 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})}) | ||
Theorem | islpidl 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 → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))) | ||
Theorem | lpi0 19068 | The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑃 = (LPIdeal‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑃) | ||
Theorem | lpi1 19069 | The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑃 = (LPIdeal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑃) | ||
Theorem | islpir 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)) | ||
Theorem | lpiss 19071 | Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ 𝑃 = (LPIdeal‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝑈) | ||
Theorem | islpir2 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 ∧ 𝑈 ⊆ 𝑃)) | ||
Theorem | lpirring 19073 | Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring) | ||
Theorem | drnglpir 19074 | Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR) | ||
Theorem | rspsn 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) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) | ||
Theorem | lidldvgen 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) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥))) | ||
Theorem | lpigen 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) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐼 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 𝑥 ∥ 𝑦)) | ||
Syntax | cnzr 19078 | The class of nonzero rings. |
class NzRing | ||
Definition | df-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 ∣ (1r‘𝑟) ≠ (0g‘𝑟)} | ||
Theorem | isnzr 19080 | Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) | ||
Theorem | nzrnz 19081 | One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) | ||
Theorem | nzrring 19082 | A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | ||
Theorem | drngnzr 19083 | All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | ||
Theorem | isnzr2 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 ∧ 2𝑜 ≼ 𝐵)) | ||
Theorem | isnzr2hash 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))) | ||
Theorem | opprnzr 19086 | The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) | ||
Theorem | ringelnzr 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 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing) | ||
Theorem | nzrunit 19088 | A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ≠ 0 ) | ||
Theorem | subrgnzr 19089 | A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing) | ||
Theorem | 0ringnnzr 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)) | ||
Theorem | 0ring 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 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 𝐵 = { 0 }) | ||
Theorem | 0ring01eq 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 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 0 = 1 ) | ||
Theorem | 01eq0ring 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 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) | ||
Theorem | 0ring01eqbi 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 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐵 ≈ 1𝑜 ↔ 1 = 0 )) | ||
Theorem | rng1nnzr 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〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) | ||
Theorem | ring1zr 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) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | rngen1zr 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) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1𝑜 ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | ringen1zr 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) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1𝑜 ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | rng1nfld 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〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ Field) | ||
Syntax | crlreg 19100 | Set of left-regular elements in a ring. |
class RLReg |
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