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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rhmf 20501 | A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶) | ||
Theorem | rhmmul 20502 | A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))) | ||
Theorem | isrhm2d 20503* | Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (1r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | isrhmd 20504* | Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (1r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ + = (+g‘𝑅) & ⊢ ⨣ = (+g‘𝑆) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | rhm1 20505 | Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (1r‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 1 ) = 𝑁) | ||
Theorem | idrhm 20506 | The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅)) | ||
Theorem | rhmf1o 20507 | A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RingHom 𝑅))) | ||
Theorem | isrim 20508 | An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) | ||
Theorem | isrimOLD 20509 | Obsolete version of isrim 20508 as of 12-Jan-2025. (Contributed by AV, 22-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))) | ||
Theorem | rimf1o 20510 | An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) | ||
Theorem | rimrhmOLD 20511 | Obsolete version of rimrhm 20512 as of 12-Jan-2025. (Contributed by AV, 22-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | rimrhm 20512 | A ring isomorphism is a homomorphism. Compare gimghm 19294. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | rimgim 20513 | An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆)) | ||
Theorem | rimisrngim 20514 | Each unital ring isomorphism is a non-unital ring isomorphism. (Contributed by AV, 30-Mar-2025.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RngIso 𝑆)) | ||
Theorem | rhmfn 20515 | The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.) |
⊢ RingHom Fn (Ring × Ring) | ||
Theorem | rhmval 20516 | The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.) |
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) | ||
Theorem | rhmco 20517 | The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ ((𝐹 ∈ (𝑇 RingHom 𝑈) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RingHom 𝑈)) | ||
Theorem | pwsco1rhm 20518* | Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐴) & ⊢ 𝑍 = (𝑅 ↑s 𝐵) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 RingHom 𝑌)) | ||
Theorem | pwsco2rhm 20519* | Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐴) & ⊢ 𝑍 = (𝑆 ↑s 𝐴) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) ⇒ ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 RingHom 𝑍)) | ||
Theorem | brric 20520 | The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) | ||
Theorem | brrici 20521 | Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃𝑟 𝑆) | ||
Theorem | brric2 20522* | The relation "is isomorphic to" for (unital) rings. This theorem corresponds to Definition df-risc 37969 of the ring isomorphism relation in JM's mathbox. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆))) | ||
Theorem | ricgic 20523 | If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ≃𝑔 𝑆) | ||
Theorem | rhmdvdsr 20524 | A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ / = (∥r‘𝑆) ⇒ ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵)) | ||
Theorem | rhmopp 20525 | A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆))) | ||
Theorem | elrhmunit 20526 | Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆)) | ||
Theorem | rhmunitinv 20527 | Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴))) | ||
Syntax | cnzr 20528 | The class of nonzero rings. |
class NzRing | ||
Definition | df-nzr 20529 | 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 20530 | 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 20531 | 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 20532 | A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.) |
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | ||
Theorem | nzrringOLD 20533 | Obsolete version of nzrring 20532 as of 23-Feb-2025. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | ||
Theorem | isnzr2 20534 | 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 20535 | Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20534. (Contributed by AV, 14-Apr-2019.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵))) | ||
Theorem | nzrpropd 20536* | If two structures have the same components (properties), one is a nonzero ring iff the other one is. (Contributed by SN, 21-Jun-2025.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ NzRing ↔ 𝐿 ∈ NzRing)) | ||
Theorem | opprnzrb 20537 | The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 20538. (Contributed by SN, 20-Jun-2025.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing) | ||
Theorem | opprnzr 20538 | The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) | ||
Theorem | ringelnzr 20539 | 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 20540 | 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 | 0ringnnzr 20541 | 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 20542 | 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 | 0ringdif 20543 | A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 })) | ||
Theorem | 0ringbas 20544 | The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 𝐵 = { 0 }) | ||
Theorem | 0ring01eq 20545 | 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 20546 | 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.) (Proof shortened by SN, 23-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) | ||
Theorem | 01eq0ringOLD 20547 | Obsolete version of 01eq0ring 20546 as of 23-Feb-2025. (Contributed by AV, 16-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) | ||
Theorem | 0ring01eqbi 20548 | 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 | 0ring1eq0 20549 | In a zero ring, a ring which is not a nonzero ring, the ring unity equals the zero element. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 1 = 0 ) | ||
Theorem | c0rhm 20550* | The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Ring ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RingHom 𝑇)) | ||
Theorem | c0rnghm 20551* | The constant mapping to zero is a non-unital ring homomorphism from any non-unital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHom 𝑇)) | ||
Theorem | zrrnghm 20552* | The constant mapping to zero is a non-unital ring homomorphism from the zero ring to any non-unital ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 𝑆)) | ||
Theorem | nrhmzr 20553 | There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.) |
⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅) | ||
Syntax | clring 20554 | Extend class notation with class of all local rings. |
class LRing | ||
Definition | df-lring 20555* | A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))} | ||
Theorem | islring 20556* | The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))) | ||
Theorem | lringnzr 20557 | A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.) |
⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | ||
Theorem | lringring 20558 | A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring) | ||
Theorem | lringnz 20559 | A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ LRing → 1 ≠ 0 ) | ||
Theorem | lringuplu 20560 | If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑈 = (Unit‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ LRing) & ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)) | ||
Syntax | csubrng 20561 | Extend class notation with all subrings of a non-unital ring. |
class SubRng | ||
Definition | df-subrng 20562* | Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.) |
⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng}) | ||
Theorem | issubrng 20563 | The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)) | ||
Theorem | subrngss 20564 | A subring is a subset. (Contributed by AV, 14-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵) | ||
Theorem | subrngid 20565 | Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅)) | ||
Theorem | subrngrng 20566 | A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng) | ||
Theorem | subrngrcl 20567 | Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.) |
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng) | ||
Theorem | subrngsubg 20568 | A subring is a subgroup. (Contributed by AV, 14-Feb-2025.) |
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | ||
Theorem | subrngringnsg 20569 | A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.) |
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅)) | ||
Theorem | subrngbas 20570 | Base set of a subring structure. (Contributed by AV, 14-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆)) | ||
Theorem | subrng0 20571 | A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 0 = (0g‘𝑆)) | ||
Theorem | subrngacl 20572 | A subring is closed under addition. (Contributed by AV, 14-Feb-2025.) |
⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴) | ||
Theorem | subrngmcl 20573 | A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 20600. (Revised by AV, 14-Feb-2025.) |
⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) | ||
Theorem | issubrng2 20574* | Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))) | ||
Theorem | opprsubrng 20575 | Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubRng‘𝑅) = (SubRng‘𝑂) | ||
Theorem | subrngint 20576 | The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.) |
⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRng‘𝑅)) | ||
Theorem | subrngin 20577 | The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.) |
⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRng‘𝑅)) | ||
Theorem | subrngmre 20578 | The subrings of a non-unital ring are a Moore system. (Contributed by AV, 15-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Rng → (SubRng‘𝑅) ∈ (Moore‘𝐵)) | ||
Theorem | subsubrng 20579 | A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴))) | ||
Theorem | subsubrng2 20580 | The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴)) | ||
Theorem | rhmimasubrnglem 20581* | Lemma for rhmimasubrng 20582: Modified part of mhmima 18850. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.) |
⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)) | ||
Theorem | rhmimasubrng 20582 | The homomorphic image of a subring is a subring. (Contributed by AV, 16-Feb-2025.) |
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁)) | ||
Theorem | cntzsubrng 20583 | Centralizers in a non-unital ring are subrings. (Contributed by AV, 17-Feb-2025.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRng‘𝑅)) | ||
Theorem | subrngpropd 20584* | If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿)) | ||
Syntax | csubrg 20585 | Extend class notation with all subrings of a ring. |
class SubRing | ||
Definition | df-subrg 20586* |
Define a subring of a ring as a set of elements that is a ring in its
own right and contains the multiplicative identity.
The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity 〈1, 0〉 which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)}) | ||
Theorem | issubrg 20587 | The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))) | ||
Theorem | subrgss 20588 | A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵) | ||
Theorem | subrgid 20589 | Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) | ||
Theorem | subrgring 20590 | A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) | ||
Theorem | subrgcrng 20591 | A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ CRing) | ||
Theorem | subrgrcl 20592 | Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | ||
Theorem | subrgsubg 20593 | A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | ||
Theorem | subrgsubrng 20594 | A subring of a unital ring is a subring of a non-unital ring. (Contributed by AV, 30-Mar-2025.) |
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubRng‘𝑅)) | ||
Theorem | subrg0 20595 | A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆)) | ||
Theorem | subrg1cl 20596 | A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴) | ||
Theorem | subrgbas 20597 | Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) | ||
Theorem | subrg1 20598 | A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆)) | ||
Theorem | subrgacl 20599 | A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴) | ||
Theorem | subrgmcl 20600 | A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 30-Mar-2025.) |
⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) |
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