P. 215 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2idlval 21401	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ 𝑇 = (𝐼 ∩ 𝐽)
Theorem	isridl 21402*	A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
Theorem	2idlelb 21403	Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21422. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ 𝐽 = (LIdeal‘𝑂)    &   ⊢ 𝑇 = (2Ideal‘𝑅)    ⇒   ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))
Theorem	2idllidld 21404	A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
Theorem	2idlridld 21405	A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑂))
Theorem	df2idl2rng 21406*	Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼)))
Theorem	df2idl2 21407*	Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
Theorem	ridl0 21408	Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
Theorem	ridl1 21409	Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
Theorem	2idl0 21410	Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝐼)
Theorem	2idl1 21411	Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝐼)
Theorem	2idlss 21412	A two-sided ideal is a subset of the base set. Formerly part of proof for 2idlcpbl 21422. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐼 = (2Ideal‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
Theorem	2idlbas 21413	The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 𝐵 = (Base‘𝐽)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐼)
Theorem	2idlelbas 21414	The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 𝐵 = (Base‘𝐽)    ⇒   ⊢ (𝜑 → (𝐵 ∈ (LIdeal‘𝑅) ∧ 𝐵 ∈ (LIdeal‘(oppr‘𝑅))))
Theorem	rng2idlsubrng 21415	A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
Theorem	rng2idlnsg 21416	A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	rng2idl0 21417	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
Theorem	rng2idlsubgsubrng 21418	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
Theorem	rng2idlsubgnsg 21419	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	rng2idlsubg0 21420	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
Theorem	2idlcpblrng 21421	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	2idlcpbl 21422	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	qus2idrng 21423	The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 21425 analog). (Contributed by AV, 23-Feb-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
Theorem	qus1 21424	The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈)))
Theorem	qusring 21425	If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
Theorem	qusrhm 21426*	If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))
Theorem	rhmpreimaidl 21427	The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)
Theorem	kerlidl 21428	The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ { 0 }) ∈ 𝐼)
Theorem	qusmul2idl 21429	Value of the ring operation in a quotient ring by a two-sided ideal. (Contributed by Thierry Arnoux, 1-Sep-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
Theorem	crngridl 21430	In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))
Theorem	crng2idl 21431	In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
Theorem	df2idl2crng 21432*	The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ 𝑈 = (2Ideal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
Theorem	qusmulrng 21433	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21434. Similar to qusmul2idl 21429. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
⊢ ∼ = (𝑅 ~QG 𝑆)    &   ⊢ 𝐻 = (𝑅 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝐻)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	quscrng 21434	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
Theorem	qusmulcrng 21435	Value of the ring operation in a quotient ring of a commutative ring. (Contributed by Thierry Arnoux, 1-Sep-2024.) (Proof shortened by metakunt, 3-Jun-2025.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
Theorem	rhmqusnsg 21436*	The mapping 𝐽 induced by a ring homomorphism 𝐹 from a subring 𝑁 of the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 13-May-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝐺 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
10.7.3.1  Condition for a non-unital ring to be unital In MathOverflow, the following theorem is claimed: "Theorem 1. Let A be a rng (= nonunital associative ring). Let J be a (two-sided) ideal of A. Assume that both rngs J and A/J are unital. Then, the rng A is also unital.", see https://mathoverflow.net/questions/487676 (/unitality-of-rngs-is-inherited-by-extensions). This thread also contains some hints to literature: Clifford and Preston's book "The Algebraic Theory of Semigroups"(Chapter 5 on representation theory), and Okninski's book Semigroup Algebras, Corollary 8 in Chapter 4. In the following, this theorem is proven formally, see rngringbdlem2 21458 (and variants rngringbd 21459 and ring2idlqusb 21461). This theorem is not trivial, since it is possible for a subset of a ring, especially a subring of a non-unital ring or (left/two-sided) ideal, to be a unital ring with a different ring unity. See also the comment for df-subrg 20680. In the given case, however, the ring unity of the larger ring can be determined by the ring unity of the two-sided ideal and a representative of the ring unity of the corresponding quotient, see ring2idlqus1 21470. An example for such a construction is given in pzriprng1ALT 21657, for the case mentioned in the comment for df-subrg 20680.
Theorem	rngqiprng1elbas 21437	The ring unity of a two-sided ideal of a non-unital ring belongs to the base set of the ring. (Contributed by AV, 15-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ (𝜑 → 1 ∈ 𝐵)
Theorem	rngqiprngghmlem1 21438	Lemma 1 for rngqiprngghm 21450. (Contributed by AV, 25-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ (Base‘𝐽))
Theorem	rngqiprngghmlem2 21439	Lemma 2 for rngqiprngghm 21450. (Contributed by AV, 25-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴)(+g‘𝐽)( 1 · 𝐶)) ∈ (Base‘𝐽))
Theorem	rngqiprngghmlem3 21440	Lemma 3 for rngqiprngghm 21450. (Contributed by AV, 25-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · (𝐴(+g‘𝑅)𝐶)) = (( 1 · 𝐴)(+g‘𝐽)( 1 · 𝐶)))
Theorem	rngqiprngimfolem 21441	Lemma for rngqiprngimfo 21452. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐼 ∧ 𝐶 ∈ 𝐵) → ( 1 · ((𝐶(-g‘𝑅)( 1 · 𝐶))(+g‘𝑅)𝐴)) = 𝐴)
Theorem	rngqiprnglinlem1 21442	Lemma 1 for rngqiprnglin 21453. (Contributed by AV, 28-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · ( 1 · 𝐶)) = ( 1 · (𝐴 · 𝐶)))
Theorem	rngqiprnglinlem2 21443	Lemma 2 for rngqiprnglin 21453. (Contributed by AV, 28-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴 · 𝐶)] ∼ = ([𝐴] ∼ (.r‘𝑄)[𝐶] ∼ ))
Theorem	rngqiprnglinlem3 21444	Lemma 3 for rngqiprnglin 21453. (Contributed by AV, 28-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (.r‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
Theorem	rngqiprngimf1lem 21445	Lemma for rngqiprngimf1 21451. (Contributed by AV, 7-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (([𝐴] ∼ = (0g‘𝑄) ∧ ( 1 · 𝐴) = (0g‘𝐽)) → 𝐴 = (0g‘𝑅)))
Theorem	rngqipbas 21446	The base set of the product of the quotient with a two-sided ideal and the two-sided ideal is the cartesian product of the base set of the quotient and the base set of the two-sided ideal. (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    ⇒   ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
Theorem	rngqiprng 21447	The product of the quotient with a two-sided ideal and the two-sided ideal is a non-unital ring. (Contributed by AV, 23-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    ⇒   ⊢ (𝜑 → 𝑃 ∈ Rng)
Theorem	rngqiprngimf 21448*	𝐹 is a function from (the base set of) a non-unital ring to the product of the (base set 𝐶 of the) quotient with a two-sided ideal and the (base set 𝐼 of the) two-sided ideal (because of 2idlbas 21413, (Base‘𝐽) = 𝐼!) (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
Theorem	rngqiprngimfv 21449*	The value of the function 𝐹 at an element of (the base set of) a non-unital ring. (Contributed by AV, 24-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
Theorem	rngqiprngghm 21450*	𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))
Theorem	rngqiprngimf1 21451*	𝐹 is a one-to-one function from (the base set of) a non-unital ring to the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 7-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))
Theorem	rngqiprngimfo 21452*	𝐹 is a function from (the base set of) a non-unital ring onto the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹:𝐵–onto→(𝐶 × 𝐼))
Theorem	rngqiprnglin 21453*	𝐹 is linear with respect to the multiplication. (Contributed by AV, 28-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))
Theorem	rngqiprngho 21454*	𝐹 is a homomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑃))
Theorem	rngqiprngim 21455*	𝐹 is an isomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngIso 𝑃))
Theorem	rng2idl1cntr 21456	The unity of a two-sided ideal of a non-unital ring is central, i.e., an element of the center of the multiplicative semigroup of the non-unital ring. This is part of the proof given in MathOverflow, which seems to be sufficient to show that 𝐹 given below (see rngqiprngimf 21448) is an isomorphism. In our proof, however we show that 𝐹 is linear regarding the multiplication (rngqiprnglin 21453) via rngqiprnglinlem1 21442 instead. (Contributed by AV, 13-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝜑 → 1 ∈ (Cntr‘𝑀))
Theorem	rngringbdlem1 21457	In a unital ring, the quotient of the ring and a two-sided ideal is unital. (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝜑 ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
Theorem	rngringbdlem2 21458	A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 14-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝜑 ∧ 𝑄 ∈ Ring) → 𝑅 ∈ Ring)
Theorem	rngringbd 21459	A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ (𝜑 → (𝑅 ∈ Ring ↔ 𝑄 ∈ Ring))
Theorem	ring2idlqus 21460*	For every unital ring there is a (two-sided) ideal of the ring (in fact the base set of the ring itself) which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 13-Feb-2025.)
⊢ (𝑅 ∈ Ring → ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring))
Theorem	ring2idlqusb 21461*	A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
⊢ (𝑅 ∈ Rng → (𝑅 ∈ Ring ↔ ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring)))
Theorem	rngqiprngfulem1 21462*	Lemma 1 for rngqiprngfu 21468 (and lemma for rngqiprngu 21469). (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ )
Theorem	rngqiprngfulem2 21463	Lemma 2 for rngqiprngfu 21468 (and lemma for rngqiprngu 21469). (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    ⇒   ⊢ (𝜑 → 𝐸 ∈ 𝐵)
Theorem	rngqiprngfulem3 21464	Lemma 3 for rngqiprngfu 21468 (and lemma for rngqiprngu 21469). (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝐵)
Theorem	rngqiprngfulem4 21465	Lemma 4 for rngqiprngfu 21468. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )
Theorem	rngqiprngfulem5 21466	Lemma 5 for rngqiprngfu 21468. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → ( 1 · 𝑈) = 1 )
Theorem	rngqipring1 21467	The ring unity of the product of the quotient with a two-sided ideal and the two-sided ideal, which both are rings. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    &   ⊢ 𝑃 = (𝑄 ×s 𝐽)    ⇒   ⊢ (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)
Theorem	rngqiprngfu 21468*	The function value of 𝐹 at the constructed expected ring unity of 𝑅 is the ring unity of the product of the quotient with the two-sided ideal and the two-sided ideal. (Contributed by AV, 16-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)
Theorem	rngqiprngu 21469	If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ (𝜑 → 𝐽 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    &   ⊢ (𝜑 → 𝑄 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   ⊢ (𝜑 → (1r‘𝑅) = 𝑈)
Theorem	ring2idlqus1 21470	If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘(𝑅 ↾s 𝐼))    &   ⊢ − = (-g‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅)) ∧ ((𝑅 ↾s 𝐼) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝐼)) ∈ Ring) ∧ 𝑈 ∈ (1r‘(𝑅 /s (𝑅 ~QG 𝐼)))) → (𝑅 ∈ Ring ∧ (1r‘𝑅) = ((𝑈 − ( 1 · 𝑈)) + 1 )))
10.7.3.2  Prime Ideals
Syntax	cprmidl 21471	Extend class notation with the class of prime ideals.
class PrmIdeal
Definition	df-prmidl 21472*	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 prmidl2 21477 and isprmidlc 21483. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
⊢ PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	prmidlval 21473*	The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	isprmidl 21474*	The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
Theorem	prmidlnr 21475	A prime ideal is a proper 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)) → 𝑃 ≠ 𝐵)
Theorem	prmidl 21476*	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 21477*	A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38749 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 21478	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 21479	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 21480	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 21481	Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
Theorem	cringm4 21482	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 21483*	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 21484	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	prmidlc2 21485	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → 𝐽 ∈ 𝑃)
Theorem	cmprmidlmcl 21486	The complement of a prime ideal is multiplicatively closed. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ (𝐵 ∖ 𝑃))) → (𝐼 · 𝐽) ∈ (𝐵 ∖ 𝑃))
Theorem	prmidlprop 21487	Property of prime ideals. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑃 ∨ 𝑌 ∈ 𝑃))
Theorem	0ringprmidl 21488	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 21489	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 21490	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 21491	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 21492	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 21493	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 21494	A quotient of a nonzero ring by a proper ideal is a nonzero 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 21495*	Lemma for ssdifidl 21496: The set 𝑃 used in the proof of ssdifidl 21496 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 21496*	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 21497*	If the set 𝑆 of ssdifidl 21496 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))
Theorem	prmidlsubm 21498	The complement of a prime ideal is multiplicatively closed. Converse of ssdifidlprm 21497. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑃) ∈ (SubMnd‘(mulGrp‘𝑅)))
10.7.4  Principal ideal rings. Divisibility in the integers
Syntax	clpidl 21499	Ring left-principal-ideal function.
class LPIdeal
Syntax	clpir 21500	Class of left principal ideal rings.
class LPIR