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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | zerdivemp1x 36301* | In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19940. (Contributed by Jeff Madsen, 18-Apr-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍))) | ||
Syntax | cdrng 36302 | Extend class notation with the class of all division rings. |
class DivRingOps | ||
Definition | df-drngo 36303* | Define the class of all division rings (sometimes called skew fields). A division ring is a unital ring where every element except the additive identity has a multiplicative inverse. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.) |
⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} | ||
Theorem | isdivrngo 36304 | The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.) |
⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))) | ||
Theorem | drngoi 36305 | The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) | ||
Theorem | gidsn 36306 | Obsolete as of 23-Jan-2020. Use mnd1id 18532 instead. The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴 | ||
Theorem | zrdivrng 36307 | The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps | ||
Theorem | dvrunz 36308 | In a division ring the ring unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍) | ||
Theorem | isgrpda 36309* | Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) & ⊢ (𝜑 → 𝑈 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈𝐺𝑥) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝑋 (𝑛𝐺𝑥) = 𝑈) ⇒ ⊢ (𝜑 → 𝐺 ∈ GrpOp) | ||
Theorem | isdrngo1 36310 | The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) | ||
Theorem | divrngcl 36311 | The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍})) | ||
Theorem | isdrngo2 36312* | A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))) | ||
Theorem | isdrngo3 36313* | A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈))) | ||
Syntax | crnghom 36314 | Extend class notation with the class of ring homomorphisms. |
class RngHom | ||
Syntax | crngiso 36315 | Extend class notation with the class of ring isomorphisms. |
class RngIso | ||
Syntax | crisc 36316 | Extend class notation with the ring isomorphism relation. |
class ≃𝑟 | ||
Definition | df-rngohom 36317* | Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ RngHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1st ‘𝑠) ↑m ran (1st ‘𝑟)) ∣ ((𝑓‘(GId‘(2nd ‘𝑟))) = (GId‘(2nd ‘𝑠)) ∧ ∀𝑥 ∈ ran (1st ‘𝑟)∀𝑦 ∈ ran (1st ‘𝑟)((𝑓‘(𝑥(1st ‘𝑟)𝑦)) = ((𝑓‘𝑥)(1st ‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(2nd ‘𝑟)𝑦)) = ((𝑓‘𝑥)(2nd ‘𝑠)(𝑓‘𝑦))))}) | ||
Theorem | rngohomval 36318* | The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝐾 = (2nd ‘𝑆) & ⊢ 𝑌 = ran 𝐽 & ⊢ 𝑉 = (GId‘𝐾) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ((𝑓‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓‘𝑥)𝐾(𝑓‘𝑦))))}) | ||
Theorem | isrngohom 36319* | The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝐾 = (2nd ‘𝑆) & ⊢ 𝑌 = ran 𝐽 & ⊢ 𝑉 = (GId‘𝐾) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))) | ||
Theorem | rngohomf 36320 | A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌) | ||
Theorem | rngohomcl 36321 | Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌) | ||
Theorem | rngohom1 36322 | A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.) |
⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) & ⊢ 𝐾 = (2nd ‘𝑆) & ⊢ 𝑉 = (GId‘𝐾) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉) | ||
Theorem | rngohomadd 36323 | Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹‘𝐴)𝐽(𝐹‘𝐵))) | ||
Theorem | rngohommul 36324 | Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝐾 = (2nd ‘𝑆) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) | ||
Theorem | rngogrphom 36325 | A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐽 = (1st ‘𝑆) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) | ||
Theorem | rngohom0 36326 | A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑊 = (GId‘𝐽) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑍) = 𝑊) | ||
Theorem | rngohomsub 36327 | Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐻 = ( /𝑔 ‘𝐺) & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝐾 = ( /𝑔 ‘𝐽) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) | ||
Theorem | rngohomco 36328 | The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇)) | ||
Theorem | rngokerinj 36329 | A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = (GId‘𝐺) & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 & ⊢ 𝑍 = (GId‘𝐽) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑍}) = {𝑊})) | ||
Definition | df-rngoiso 36330* | Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)}) | ||
Theorem | rngoisoval 36331* | The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋–1-1-onto→𝑌}) | ||
Theorem | isrngoiso 36332 | The predicate "is a ring isomorphism between 𝑅 and 𝑆". (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–1-1-onto→𝑌))) | ||
Theorem | rngoiso1o 36333 | A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌) | ||
Theorem | rngoisohom 36334 | A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆)) | ||
Theorem | rngoisocnv 36335 | The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ◡𝐹 ∈ (𝑆 RngIso 𝑅)) | ||
Theorem | rngoisoco 36336 | The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RngIso 𝑇)) | ||
Definition | df-risc 36337* | Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))} | ||
Theorem | isriscg 36338* | The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) | ||
Theorem | isrisc 36339* | The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝑅 ∈ V & ⊢ 𝑆 ∈ V ⇒ ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))) | ||
Theorem | risc 36340* | The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))) | ||
Theorem | risci 36341 | Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅 ≃𝑟 𝑆) | ||
Theorem | riscer 36342 | Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ ≃𝑟 Er dom ≃𝑟 | ||
Syntax | ccm2 36343 | Extend class notation with a class that adds commutativity to various flavors of rings. |
class Com2 | ||
Definition | df-com2 36344* | A device to add commutativity to various sorts of rings. I use ran 𝑔 because I suppose 𝑔 has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.) |
⊢ Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)} | ||
Syntax | cfld 36345 | Extend class notation with the class of all fields. |
class Fld | ||
Definition | df-fld 36346 | Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) |
⊢ Fld = (DivRingOps ∩ Com2) | ||
Syntax | ccring 36347 | Extend class notation with the class of commutative rings. |
class CRingOps | ||
Definition | df-crngo 36348 | Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ CRingOps = (RingOps ∩ Com2) | ||
Theorem | iscom2 36349* | A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.) |
⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎))) | ||
Theorem | iscrngo 36350 | The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | ||
Theorem | iscrngo2 36351* | The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))) | ||
Theorem | iscringd 36352* | Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.) |
⊢ (𝜑 → 𝐺 ∈ AbelOp) & ⊢ (𝜑 → 𝑋 = ran 𝐺) & ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) & ⊢ (𝜑 → 𝑈 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ⇒ ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps) | ||
Theorem | flddivrng 36353 | A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps) | ||
Theorem | crngorngo 36354 | A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) | ||
Theorem | crngocom 36355 | The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) | ||
Theorem | crngm23 36356 | Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵)) | ||
Theorem | crngm4 36357 | Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷))) | ||
Theorem | fldcrngo 36358 | A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) | ||
Theorem | isfld2 36359 | The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) | ||
Theorem | crngohomfo 36360 | The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐽 = (1st ‘𝑆) & ⊢ 𝑌 = ran 𝐽 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋–onto→𝑌)) → 𝑆 ∈ CRingOps) | ||
Syntax | cidl 36361 | Extend class notation with the class of ideals. |
class Idl | ||
Syntax | cpridl 36362 | Extend class notation with the class of prime ideals. |
class PrIdl | ||
Syntax | cmaxidl 36363 | Extend class notation with the class of maximal ideals. |
class MaxIdl | ||
Definition | df-idl 36364* | Define the class of (two-sided) ideals of a ring 𝑅. A subset of 𝑅 is an ideal if it contains 0, is closed under addition, and is closed under multiplication on either side by any element of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st ‘𝑟) ∣ ((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))}) | ||
Definition | df-pridl 36365* | Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵 ⊆ 𝐼 for ideals 𝐴 and 𝐵, either 𝐴 ⊆ 𝐼 or 𝐵 ⊆ 𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see ispridl2 36392 and ispridlc 36424. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}) | ||
Definition | df-maxidl 36366* | Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))}) | ||
Theorem | idlval 36367* | The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))}) | ||
Theorem | isidl 36368* | The predicate "is an ideal of the ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))) | ||
Theorem | isidlc 36369* | The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 (𝑧𝐻𝑥) ∈ 𝐼)))) | ||
Theorem | idlss 36370 | An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋) | ||
Theorem | idlcl 36371 | An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋) | ||
Theorem | idl0cl 36372 | An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍 ∈ 𝐼) | ||
Theorem | idladdcl 36373 | An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼) | ||
Theorem | idllmulcl 36374 | An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼) | ||
Theorem | idlrmulcl 36375 | An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼) | ||
Theorem | idlnegcl 36376 | An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼) | ||
Theorem | idlsubcl 36377 | An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼) | ||
Theorem | rngoidl 36378 | A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) | ||
Theorem | 0idl 36379 | The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅)) | ||
Theorem | 1idl 36380 | Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) | ||
Theorem | 0rngo 36381 | In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) | ||
Theorem | divrngidl 36382 | The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋}) | ||
Theorem | intidl 36383 | The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅)) | ||
Theorem | inidl 36384 | The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅)) | ||
Theorem | unichnidl 36385* | The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (Idl‘𝑅)) | ||
Theorem | keridl 36386 | The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑆) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅)) | ||
Theorem | pridlval 36387* | The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}) | ||
Theorem | ispridl 36388* | The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))) | ||
Theorem | pridlidl 36389 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅)) | ||
Theorem | pridlnr 36390 | A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ≠ 𝑋) | ||
Theorem | pridl 36391* | The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐻 = (2nd ‘𝑅) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃)) | ||
Theorem | ispridl2 36392* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 36424 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅)) | ||
Theorem | maxidlval 36393* | The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))}) | ||
Theorem | ismaxidl 36394* | The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) | ||
Theorem | maxidlidl 36395 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅)) | ||
Theorem | maxidlnr 36396 | A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋) | ||
Theorem | maxidlmax 36397 | A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋)) | ||
Theorem | maxidln1 36398 | One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) |
⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) | ||
Theorem | maxidln0 36399 | A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) | ||
Syntax | cprrng 36400 | Extend class notation with the class of prime rings. |
class PrRing |
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