HomeTheorem List (p. 335 of 497)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | prmidl 33401* | The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) | ||
| Theorem | prmidl2 33402* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38040 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | idlmulssprm 33403 | Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
| ⊢ × = (LSSum‘(mulGrp‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃) ⇒ ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) | ||
| Theorem | pridln1 33404 | A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼) | ||
| Theorem | prmidlidl 33405 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅)) | ||
| Theorem | prmidlssidl 33406 | Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) | ||
| Theorem | cringm4 33407 | Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊))) | ||
| Theorem | isprmidlc 33408* | The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))))) | ||
| Theorem | prmidlc 33409 | Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃)) | ||
| Theorem | 0ringprmidl 33410 | The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅) | ||
| Theorem | prmidl0 33411 | The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
| ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn) | ||
| Theorem | rhmpreimaprmidl 33412 | The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.) |
| ⊢ 𝑃 = (PrmIdeal‘𝑅) ⇒ ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃) | ||
| Theorem | qsidomlem1 33413 | If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | qsidomlem2 33414 | A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn) | ||
| Theorem | qsidom 33415 | An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅))) | ||
| Theorem | qsnzr 33416 | A quotient of a non-zero ring by a proper ideal is a non-zero ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) & ⊢ (𝜑 → 𝐼 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝑄 ∈ NzRing) | ||
| Theorem | ssdifidllem 33417* | Lemma for ssdifidl 33418: The set 𝑃 used in the proof of ssdifidl 33418 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅) & ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)} & ⊢ (𝜑 → 𝑍 ⊆ 𝑃) & ⊢ (𝜑 → 𝑍 ≠ ∅) & ⊢ (𝜑 → [⊊] Or 𝑍) ⇒ ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃) | ||
| Theorem | ssdifidl 33418* | Let 𝑅 be a ring, and let 𝐼 be an ideal of 𝑅 disjoint with a set 𝑆. Then there exists an ideal 𝑖, maximal among the set 𝑃 of ideals containing 𝐼 and disjoint with 𝑆. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅) & ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)} ⇒ ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗) | ||
| Theorem | ssdifidlprm 33419* | If the set 𝑆 of ssdifidl 33418 is multiplicatively closed, then the ideal 𝑖 is prime. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀)) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅) & ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)} ⇒ ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗)) | ||
| Syntax | cmxidl 33420 | Extend class notation with the class of maximal ideals. |
| class MaxIdeal | ||
| Definition | df-mxidl 33421* | 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.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))}) | ||
| Theorem | mxidlval 33422* | The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))}) | ||
| Theorem | ismxidl 33423* | The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))) | ||
| Theorem | mxidlidl 33424 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) | ||
| Theorem | mxidlnr 33425 | A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) | ||
| Theorem | mxidlmax 33426 | A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)) | ||
| Theorem | mxidln1 33427 | One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1 ∈ 𝑀) | ||
| Theorem | mxidlnzr 33428 | A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing) | ||
| Theorem | mxidlmaxv 33429 | An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ⊆ 𝐼) & ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝐼 = 𝐵) | ||
| Theorem | crngmxidl 33430 | In a commutative ring, maximal ideals of the opposite ring coincide with maximal ideals. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑀 = (MaxIdeal‘𝑂)) | ||
| Theorem | mxidlprm 33431 | Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ × = (LSSum‘(mulGrp‘𝑅)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | mxidlirredi 33432 | In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (𝐾‘{𝑋}) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅)) | ||
| Theorem | mxidlirred 33433 | In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (𝐾‘{𝑋}) & ⊢ (𝜑 → 𝑅 ∈ PID) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅))) | ||
| Theorem | ssmxidllem 33434* | The set 𝑃 used in the proof of ssmxidl 33435 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ≠ 𝐵) & ⊢ (𝜑 → 𝑍 ⊆ 𝑃) & ⊢ (𝜑 → 𝑍 ≠ ∅) & ⊢ (𝜑 → [⊊] Or 𝑍) ⇒ ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃) | ||
| Theorem | ssmxidl 33435* | Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼 ⊆ 𝑚) | ||
| Theorem | drnglidl1ne0 33436 | In a nonzero ring, the zero ideal is different of the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝐵 ≠ { 0 }) | ||
| Theorem | drng0mxidl 33437 | In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
| ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) | ||
| Theorem | drngmxidl 33438 | The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
| ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }}) | ||
| Theorem | drngmxidlr 33439 | If a ring's only maximal ideal is the zero ideal, it is a division ring. See also drngmxidl 33438. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 = {{ 0 }}) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | krull 33440* | Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) | ||
| Theorem | mxidlnzrb 33441* | A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))) | ||
| Theorem | krullndrng 33442* | Krull's theorem for non-division-rings: Existence of a nonzero maximal ideal. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑚 ≠ { 0 }) | ||
| Theorem | opprabs 33443 | The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑅) & ⊢ (𝜑 → (.r‘ndx) ∈ dom 𝑅) & ⊢ (𝜑 → · Fn (𝐵 × 𝐵)) ⇒ ⊢ (𝜑 → 𝑅 = (oppr‘𝑂)) | ||
| Theorem | oppreqg 33444 | Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼)) | ||
| Theorem | opprnsg 33445 | Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂) | ||
| Theorem | opprlidlabs 33446 | The ideals of the opposite ring's opposite ring are the ideals of the original ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂))) | ||
| Theorem | oppr2idl 33447 | Two sided ideal of the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂)) | ||
| Theorem | opprmxidlabs 33448 | The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂))) | ||
| Theorem | opprqusbas 33449 | The base of the quotient of the opposite ring is the same as the base of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
| Theorem | opprqusplusg 33450 | The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) & ⊢ 𝐸 = (Base‘𝑄) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌)) | ||
| Theorem | opprqus0g 33451 | The group identity element of the quotient of the opposite ring is the same as the group identity element of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) ⇒ ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
| Theorem | opprqusmulr 33452 | The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) & ⊢ 𝐸 = (Base‘𝑄) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌)) | ||
| Theorem | opprqus1r 33453 | The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) ⇒ ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
| Theorem | opprqusdrng 33454 | The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) ⇒ ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing)) | ||
| Theorem | qsdrngilem 33455* | Lemma for qsdrngi 33456. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄)) | ||
| Theorem | qsdrngi 33456 | A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂)) ⇒ ⊢ (𝜑 → 𝑄 ∈ DivRing) | ||
| Theorem | qsdrnglem2 33457 | Lemma for qsdrng 33458. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ DivRing) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ (𝐽 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝐽 = 𝐵) | ||
| Theorem | qsdrng 33458 | An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂)))) | ||
| Theorem | qsfld 33459 | An ideal 𝑀 in the commutative ring 𝑅 is maximal if and only if the factor ring 𝑄 is a field. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅))) | ||
| Theorem | mxidlprmALT 33460 | Every maximal ideal is prime - alternative proof. (Contributed by Thierry Arnoux, 15-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑀 ∈ (PrmIdeal‘𝑅)) | ||
| Syntax | cidlsrg 33461 | Extend class notation with the semiring of ideals of a ring. |
| class IDLsrg | ||
| Definition | df-idlsrg 33462* | Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ IDLsrg = (𝑟 ∈ V ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩})) | ||
| Theorem | idlsrgstr 33463 | A constructed semiring of ideals is a structure. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}) ⇒ ⊢ 𝑊 Struct ⟨1, ;10⟩ | ||
| Theorem | idlsrgval 33464* | Lemma for idlsrgbas 33465 through idlsrgtset 33469. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ ⊕ = (LSSum‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ⊗ = (LSSum‘𝐺) ⇒ ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩})) | ||
| Theorem | idlsrgbas 33465 | Base of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐼 = (Base‘𝑆)) | ||
| Theorem | idlsrgplusg 33466 | Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ ⊕ = (LSSum‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆)) | ||
| Theorem | idlsrg0g 33467 | The zero ideal is the additive identity of the semiring of ideals. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → { 0 } = (0g‘𝑆)) | ||
| Theorem | idlsrgmulr 33468* | Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ⊗ = (LSSum‘𝐺) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗))) = (.r‘𝑆)) | ||
| Theorem | idlsrgtset 33469* | Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopSet‘𝑆)) | ||
| Theorem | idlsrgmulrval 33470 | Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ · = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽))) | ||
| Theorem | idlsrgmulrcl 33471 | Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ∈ 𝐵) | ||
| Theorem | idlsrgmulrss1 33472 | In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐼) | ||
| Theorem | idlsrgmulrss2 33473 | The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽) | ||
| Theorem | idlsrgmulrssin 33474 | In a commutative ring, the product of two ideals is a subset of their intersection. (Contributed by Thierry Arnoux, 17-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) & ⊢ 𝐵 = (LIdeal‘𝑅) & ⊢ ⊗ = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ (𝐼 ∩ 𝐽)) | ||
| Theorem | idlsrgmnd 33475 | The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd) | ||
| Theorem | idlsrgcmnd 33476 | The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑆 = (IDLsrg‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ CMnd) | ||
| Theorem | rprmval 33477* | The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))}) | ||
| Theorem | isrprm 33478* | Property for 𝑃 to be a prime element in the ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦))))) | ||
| Theorem | rprmcl 33479 | A ring prime is an element of the base set. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
| Theorem | rprmdvds 33480 | If a ring prime 𝑄 divides a product 𝑋 · 𝑌, then it divides either 𝑋 or 𝑌. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∥ (𝑋 · 𝑌)) ⇒ ⊢ (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌)) | ||
| Theorem | rprmnz 33481 | A ring prime is nonzero. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑄 ≠ 0 ) | ||
| Theorem | rprmnunit 33482 | A ring prime is not a unit. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → ¬ 𝑄 ∈ 𝑈) | ||
| Theorem | rsprprmprmidl 33483 | In a commutative ring, ideals generated by prime elements are prime ideals. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑃 ∈ (RPrime‘𝑅)) ⇒ ⊢ (𝜑 → (𝐾‘{𝑃}) ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | rsprprmprmidlb 33484 | In an integral domain, an ideal generated by a single element is a prime iff that element is prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑃 ↔ (𝐾‘{𝑋}) ∈ (PrmIdeal‘𝑅))) | ||
| Theorem | rprmndvdsr1 33485 | A ring prime element does not divide the ring multiplicative identity. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → ¬ 𝑄 ∥ 1 ) | ||
| Theorem | rprmasso 33486 | In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∥ 𝑌) & ⊢ (𝜑 → 𝑌 ∥ 𝑋) ⇒ ⊢ (𝜑 → 𝑌 ∈ 𝑃) | ||
| Theorem | rprmasso2 33487 | In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∥ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑌 ∥ 𝑋) | ||
| Theorem | rprmasso3 33488* | In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∥ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ 𝑈 (𝑡 · 𝑋) = 𝑌) | ||
| Theorem | unitmulrprm 33489 | A ring unit multiplied by a ring prime is a ring prime. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐼 · 𝑄) ∈ 𝑃) | ||
| Theorem | rprmndvdsru 33490 | A ring prime element does not divide any ring unit. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) ⇒ ⊢ (𝜑 → ¬ 𝑄 ∥ 𝑇) | ||
| Theorem | rprmirredlem 33491 | Lemma for rprmirred 33492. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑄 ≠ 0 ) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑈)) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 = (𝑋 · 𝑌)) & ⊢ (𝜑 → 𝑄 ∥ 𝑋) ⇒ ⊢ (𝜑 → 𝑌 ∈ 𝑈) | ||
| Theorem | rprmirred 33492 | In an integral domain, ring primes are irreducible. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑄 ∈ 𝐼) | ||
| Theorem | rprmirredb 33493 | In a principal ideal domain, the converse of rprmirred 33492 holds, i.e. irreducible elements are prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ PID) ⇒ ⊢ (𝜑 → 𝐼 = 𝑃) | ||
| Theorem | rprmdvdspow 33494 | If a prime element divides a ring "power", it divides the term. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝑀) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑄 ∥ (𝑁 ↑ 𝑋)) ⇒ ⊢ (𝜑 → 𝑄 ∥ 𝑋) | ||
| Theorem | rprmdvdsprod 33495* | If a prime element 𝑄 divides a product, then it divides one term. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 1 ) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑄 ∥ (𝑀 Σg 𝐹)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐹 supp 1 )𝑄 ∥ (𝐹‘𝑥)) | ||
| Theorem | 1arithidomlem1 33496* | Lemma for 1arithidom 33498. (Contributed by Thierry Arnoux, 30-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐽 = (0..^(♯‘𝐹)) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺)) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤))))) & ⊢ (𝜑 → 𝐻 ∈ Word 𝑃) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻))) & ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻))) & ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾)) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) & ⊢ (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾)) & ⊢ (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻))) & ⊢ (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩)) & ⊢ (𝜑 → 𝑁 ∈ 𝑈) & ⊢ (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻))) ⇒ ⊢ (𝜑 → ∃𝑐∃𝑑 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑐:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝑑 ∘f · (𝐹 ∘ 𝑐)))) | ||
| Theorem | 1arithidomlem2 33497* | Lemma for 1arithidom 33498: induction step. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐽 = (0..^(♯‘𝐹)) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺)) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤))))) & ⊢ (𝜑 → 𝐻 ∈ Word 𝑃) & ⊢ (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻))) & ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻))) & ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾)) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) & ⊢ (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾)) & ⊢ (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻))) & ⊢ (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩)) & ⊢ (𝜑 → 𝑁 ∈ 𝑈) & ⊢ (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻))) & ⊢ (𝜑 → 𝐷 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))) & ⊢ (𝜑 → 𝐶:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹))) & ⊢ (𝜑 → ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝐷 ∘f · (𝐹 ∘ 𝐶))) ⇒ ⊢ (𝜑 → (((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆):(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩)))–1-1-onto→(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩))) ∧ 𝐻 = (((𝐷 ++ ⟨“𝑇”⟩) ∘ ◡𝑆) ∘f · ((𝐹 ++ ⟨“𝑄”⟩) ∘ ((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆))))) | ||
| Theorem | 1arithidom 33498* | Uniqueness of prime factorizations in an integral domain 𝑅. Given two equal products 𝐹 and 𝐺 of prime elements, 𝐹 and 𝐺 are equal up to a renumbering 𝑤 and a multiplication by units 𝑢. See also 1arith 16945. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑃 = (RPrime‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐽 = (0..^(♯‘𝐹)) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑃) & ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺)) ⇒ ⊢ (𝜑 → ∃𝑤∃𝑢 ∈ (𝑈 ↑m 𝐽)(𝑤:𝐽–1-1-onto→𝐽 ∧ 𝐺 = (𝑢 ∘f · (𝐹 ∘ 𝑤)))) | ||
| Syntax | cufd 33499 | Class of unique factorization domains. |
| class UFD | ||
| Definition | df-ufd 33500* | Define the class of unique factorization domains. A unique factorization domain (UFD for short), is an integral domain such that every nonzero prime ideal contains a prime element (this is a characterization due to Irving Kaplansky). A UFD is sometimes also called a "factorial ring" following the terminology of Bourbaki. (Contributed by Mario Carneiro, 17-Feb-2015.) Exclude the 0 prime ideal. (Revised by Thierry Arnoux, 9-May-2025.) Exclude the 0 ring. (Revised by Thierry Arnoux, 14-Jun-2025.) |
| ⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅} | ||
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