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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rhmco 20501 | 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 20502* | 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 20503* | 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 20504 | The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.) |
| ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) | ||
| Theorem | brrici 20505 | Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.) |
| ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃𝑟 𝑆) | ||
| Theorem | brric2 20506* | The relation "is isomorphic to" for (unital) rings. This theorem corresponds to Definition df-risc 37990 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 20507 | 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 20508 | 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 20509 | 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 20510 | 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 20511 | 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 20512 | The class of nonzero rings. |
| class NzRing | ||
| Definition | df-nzr 20513 | 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 20514 | 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 20515 | 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 20516 | 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 20517 | Obsolete version of nzrring 20516 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 20518 | 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 20519 | Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20518. (Contributed by AV, 14-Apr-2019.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵))) | ||
| Theorem | nzrpropd 20520* | 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 20521 | The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 20522. (Contributed by SN, 20-Jun-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing) | ||
| Theorem | opprnzr 20522 | The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) | ||
| Theorem | ringelnzr 20523 | 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 20524 | 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 20525 | 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 20526 | 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 20527 | 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 20528 | 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 20529 | 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 20530 | 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 20531 | Obsolete version of 01eq0ring 20530 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 20532 | 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 20533 | 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 20534* | 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 20535* | 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 20536* | 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 20537 | 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 20538 | Extend class notation with class of all local rings. |
| class LRing | ||
| Definition | df-lring 20539* | 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 20540* | 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 20541 | A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.) |
| ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | ||
| Theorem | lringring 20542 | 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 20543 | 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 20544 | 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 20545 | Extend class notation with all subrings of a non-unital ring. |
| class SubRng | ||
| Definition | df-subrng 20546* | 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 20547 | 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 20548 | A subring is a subset. (Contributed by AV, 14-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵) | ||
| Theorem | subrngid 20549 | 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 20550 | A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng) | ||
| Theorem | subrngrcl 20551 | Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng) | ||
| Theorem | subrngsubg 20552 | A subring is a subgroup. (Contributed by AV, 14-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | ||
| Theorem | subrngringnsg 20553 | A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅)) | ||
| Theorem | subrngbas 20554 | Base set of a subring structure. (Contributed by AV, 14-Feb-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆)) | ||
| Theorem | subrng0 20555 | 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 20556 | 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 20557 | A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 20584. (Revised by AV, 14-Feb-2025.) |
| ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) | ||
| Theorem | issubrng2 20558* | 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 20559 | Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubRng‘𝑅) = (SubRng‘𝑂) | ||
| Theorem | subrngint 20560 | 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 20561 | 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 20562 | 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 20563 | 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 20564 | 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 20565* | Lemma for rhmimasubrng 20566: Modified part of mhmima 18838. (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 20566 | 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 20567 | 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 20568* | 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 20569 | Extend class notation with all subrings of a ring. |
| class SubRing | ||
| Definition | df-subrg 20570* |
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 20571 | 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 20572 | A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵) | ||
| Theorem | subrgid 20573 | 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 20574 | A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) | ||
| Theorem | subrgcrng 20575 | 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 20576 | Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.) |
| ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | ||
| Theorem | subrgsubg 20577 | A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.) |
| ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | ||
| Theorem | subrgsubrng 20578 | 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 20579 | 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 20580 | A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
| ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴) | ||
| Theorem | subrgbas 20581 | 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 20582 | 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 20583 | 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 20584 | 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) | ||
| Theorem | subrgsubm 20585 | A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀)) | ||
| Theorem | subrgdvds 20586 | If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝐸 = (∥r‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) | ||
| Theorem | subrguss 20587 | A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑉 = (Unit‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈) | ||
| Theorem | subrginv 20588 | A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝐽 = (invr‘𝑆) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋)) | ||
| Theorem | subrgdv 20589 | A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ / = (/r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝐸 = (/r‘𝑆) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌)) | ||
| Theorem | subrgunit 20590 | An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑉 = (Unit‘𝑆) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))) | ||
| Theorem | subrgugrp 20591 | The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑉 = (Unit‘𝑆) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺)) | ||
| Theorem | issubrg2 20592* | Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))) | ||
| Theorem | opprsubrg 20593 | Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) | ||
| Theorem | subrgnzr 20594 | A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing) | ||
| Theorem | subrgint 20595 | The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
| ⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅)) | ||
| Theorem | subrgin 20596 | The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
| ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRing‘𝑅)) | ||
| Theorem | subrgmre 20597 | The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (Moore‘𝐵)) | ||
| Theorem | subsubrg 20598 | A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴))) | ||
| Theorem | subsubrg2 20599 | The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = ((SubRing‘𝑅) ∩ 𝒫 𝐴)) | ||
| Theorem | issubrg3 20600 | A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀)))) | ||
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