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Type | Label | Description |
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Statement | ||
Definition | df-rngohom 35401* | 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 35402* | 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 35403* | 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 35404 | 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 35405 | 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 35406 | 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 35407 | 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 35408 | 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 35409 | 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 35410 | 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 35411 | 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 35412 | 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 35413 | 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 35414* | 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 35415* | 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 35416 | 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 35417 | 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 35418 | 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 35419 | 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 35420 | 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 35421* | 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 35422* | 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 35423* | 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 35424* | 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 35425 | 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 35426 | 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 35427 | Extend class notation with a class that adds commutativity to various flavors of rings. |
class Com2 | ||
Definition | df-com2 35428* | 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 35429 | Extend class notation with the class of all fields. |
class Fld | ||
Definition | df-fld 35430 | 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 35431 | Extend class notation with the class of commutative rings. |
class CRingOps | ||
Definition | df-crngo 35432 | Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ CRingOps = (RingOps ∩ Com2) | ||
Theorem | iscom2 35433* | 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 35434 | The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | ||
Theorem | iscrngo2 35435* | 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 35436* | 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 35437 | 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 35438 | A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) | ||
Theorem | crngocom 35439 | 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 35440 | 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 35441 | 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 | fldcrng 35442 | A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) |
⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps) | ||
Theorem | isfld2 35443 | The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps)) | ||
Theorem | crngohomfo 35444 | 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 35445 | Extend class notation with the class of ideals. |
class Idl | ||
Syntax | cpridl 35446 | Extend class notation with the class of prime ideals. |
class PrIdl | ||
Syntax | cmaxidl 35447 | Extend class notation with the class of maximal ideals. |
class MaxIdl | ||
Definition | df-idl 35448* | 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 35449* | 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 35476 and ispridlc 35508. (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 35450* | 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 35451* | 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 35452* | 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 35453* | 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 35454 | 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 35455 | 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 35456 | 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 35457 | 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 35458 | 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 35459 | 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 35460 | 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 35461 | 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 35462 | 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 35463 | 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 35464 | 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 35465 | 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 35466 | 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 35467 | 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 35468 | 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 35469* | 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 35470 | 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 35471* | 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 35472* | 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 35473 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅)) | ||
Theorem | pridlnr 35474 | 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 35475* | 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 35476* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 35508 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 35477* | 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 35478* | 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 35479 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅)) | ||
Theorem | maxidlnr 35480 | 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 35481 | 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 35482 | 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 35483 | 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 35484 | Extend class notation with the class of prime rings. |
class PrRing | ||
Syntax | cdmn 35485 | Extend class notation with the class of domains. |
class Dmn | ||
Definition | df-prrngo 35486 | Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} | ||
Definition | df-dmn 35487 | Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ Dmn = (PrRing ∩ Com2) | ||
Theorem | isprrngo 35488 | The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) | ||
Theorem | prrngorngo 35489 | A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) | ||
Theorem | smprngopr 35490 | A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing) | ||
Theorem | divrngpr 35491 | A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) | ||
Theorem | isdmn 35492 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) | ||
Theorem | isdmn2 35493 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) | ||
Theorem | dmncrng 35494 | A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps) | ||
Theorem | dmnrngo 35495 | A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps) | ||
Theorem | flddmn 35496 | A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) | ||
Syntax | cigen 35497 | Extend class notation with the ideal generation function. |
class IdlGen | ||
Definition | df-igen 35498* | Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗}) | ||
Theorem | igenval 35499* | The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) | ||
Theorem | igenss 35500 | A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆)) |
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