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Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremellpi 33401 Elementhood in a left principal ideal in terms of the "divides" relation. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ 𝑋 𝑌))
 
Theoremlpirlidllpi 33402* In a principal ideal ring, ideals are principal. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ LPIR)    &   (𝜑𝐽𝐼)       (𝜑 → ∃𝑥𝐵 𝐽 = (𝐾‘{𝑥}))
 
Theoremrspidlid 33403 The ideal span of an ideal is the ideal itself. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐾 = (RSpan‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐾𝐼) = 𝐼)
 
Theorempidlnz 33404 A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑋0 ) → (𝐾‘{𝑋}) ≠ { 0 })
 
Theoremlbslsp 33405* Any element of a left module 𝑀 can be expressed as a linear combination of the elements of a basis 𝑉 of 𝑀. (Contributed by Thierry Arnoux, 3-Aug-2023.)
𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝑉 ∈ (LBasis‘𝑀))       (𝜑 → (𝑋𝐵 ↔ ∃𝑎 ∈ (𝐾m 𝑉)(𝑎 finSupp 0𝑋 = (𝑀 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))))))
 
Theoremlindssn 33406 Any singleton of a nonzero element is an independent set. (Contributed by Thierry Arnoux, 5-Aug-2023.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LVec ∧ 𝑋𝐵𝑋0 ) → {𝑋} ∈ (LIndS‘𝑊))
 
Theoremlindflbs 33407 Conditions for an independent family to be a basis. (Contributed by Thierry Arnoux, 21-Jul-2023.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑆 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝑆)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑆 ∈ NzRing)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼1-1𝐵)       (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵)))
 
Theoremislbs5 33408* An equivalent formulation of the basis predicate in a vector space, using a function 𝐹 for generating the base. (Contributed by Thierry Arnoux, 20-Feb-2025.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝑆)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑆 ∈ NzRing)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼1-1𝐵)       (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (∀𝑎 ∈ (𝐾m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵)))
 
Theoremlinds2eq 33409 Deduce equality of elements in an independent set. (Contributed by Thierry Arnoux, 18-Jul-2023.)
𝐹 = (Base‘(Scalar‘𝑊))    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g‘(Scalar‘𝑊))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐵 ∈ (LIndS‘𝑊))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐾𝐹)    &   (𝜑𝐿𝐹)    &   (𝜑𝐾0 )    &   (𝜑 → (𝐾 · 𝑋) = (𝐿 · 𝑌))       (𝜑 → (𝑋 = 𝑌𝐾 = 𝐿))
 
Theoremlindfpropd 33410* Property deduction for linearly independent families. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))    &   (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)    &   (𝜑𝑋𝐴)       (𝜑 → (𝑋 LIndF 𝐾𝑋 LIndF 𝐿))
 
Theoremlindspropd 33411* Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))    &   (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)       (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))
 
21.3.10.36  Ring associates, ring units
 
Theoremdvdsruassoi 33412 If two elements 𝑋 and 𝑌 of a ring 𝑅 are unit multiples, then they are associates, i.e. each divides the other. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑉𝑈)    &   (𝜑 → (𝑉 · 𝑋) = 𝑌)       (𝜑 → (𝑋 𝑌𝑌 𝑋))
 
Theoremdvdsruasso 33413* Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff they are unit multiples of each other. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ IDomn)       (𝜑 → ((𝑋 𝑌𝑌 𝑋) ↔ ∃𝑢𝑈 (𝑢 · 𝑋) = 𝑌))
 
Theoremdvdsruasso2 33414* A reformulation of dvdsruasso 33413. (Proposed by Gerard Lang, 28-May-2025.) (Contributed by Thiery Arnoux, 29-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ IDomn)    &    1 = (1r𝑅)       (𝜑 → ((𝑋 𝑌𝑌 𝑋) ↔ ∃𝑢𝑈𝑣𝑈 ((𝑢 · 𝑋) = 𝑌 ∧ (𝑣 · 𝑌) = 𝑋 ∧ (𝑢 · 𝑣) = 1 )))
 
Theoremdvdsrspss 33415 In a ring, an element 𝑋 divides 𝑌 iff the ideal generated by 𝑌 is a subset of the ideal generated by 𝑋 (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑋 𝑌 ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})))
 
Theoremrspsnasso 33416 Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff the ideals they generate are equal. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → ((𝑋 𝑌𝑌 𝑋) ↔ (𝐾‘{𝑌}) = (𝐾‘{𝑋})))
 
Theoremunitprodclb 33417 A finite product is a unit iff all factors are units. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐹 ∈ Word 𝐵)       (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹𝑈))
 
21.3.10.37  Subgroup sum / Sumset / Minkowski sum

The sumset (also called the Minkowski sum) of two subsets 𝐴 and 𝐵, is defined to be the set of all sums of an element from 𝐴 with an element from 𝐵.

The sumset operation can be used for both group (additive) operations and ring (multiplicative) operations.

 
Theoremelgrplsmsn 33418* Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))
 
Theoremlsmsnorb 33419* The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19325. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴 {𝑋}) = [𝑋] )
 
Theoremlsmsnorb2 33420* The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19325. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑥 + 𝑔) = 𝑦)}    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → ({𝑋} 𝐴) = [𝑋] )
 
Theoremelringlsm 33421* Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥𝐸𝑦𝐹 𝑍 = (𝑥 · 𝑦)))
 
Theoremelringlsmd 33422 Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐹)       (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹))
 
Theoremringlsmss 33423 Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐸 × 𝐹) ⊆ 𝐵)
 
Theoremringlsmss1 33424 The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐸𝐵)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼)
 
Theoremringlsmss2 33425 The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼)
 
Theoremlsmsnpridl 33426 The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐵 × {𝑋}) = (𝐾‘{𝑋}))
 
Theoremlsmsnidl 33427 The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐵 × {𝑋}) ∈ (LPIdeal‘𝑅))
 
Theoremlsmidllsp 33428 The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 𝐽) = (𝐾‘(𝐼𝐽)))
 
Theoremlsmidl 33429 The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 𝐽) ∈ (LIdeal‘𝑅))
 
Theoremlsmssass 33430 Group sum is associative, subset version (see lsmass 19687). (Contributed by Thierry Arnoux, 1-Jun-2024.)
= (LSSum‘𝐺)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑅𝐵)    &   (𝜑𝑇𝐵)    &   (𝜑𝑈𝐵)       (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))
 
Theoremgrplsm0l 33431 Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝐵𝐴 ≠ ∅) → ({ 0 } 𝐴) = 𝐴)
 
Theoremgrplsmid 33432 The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
= (LSSum‘𝐺)       ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)
 
21.3.10.38  The quotient map
 
Theoremquslsm 33433 Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋𝐵)       (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))
 
Theoremqusbas2 33434* Alternate definition of the group quotient set, as the set of all cosets of the form ({𝑥} 𝑁). (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &   ((𝜑𝑥𝐵) → 𝑁 ∈ (SubGrp‘𝐺))       (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = ran (𝑥𝐵 ↦ ({𝑥} 𝑁)))
 
Theoremqus0g 33435 The identity element of a quotient group. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))       (𝑁 ∈ (NrmSGrp‘𝐺) → (0g𝑄) = 𝑁)
 
Theoremqusima 33436* The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑥𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑𝐻𝑆)    &   (𝜑𝑆 ⊆ (SubGrp‘𝐺))       (𝜑 → (𝐸𝐻) = (𝐹𝐻))
 
Theoremqusrn 33437* The natural map from elements to their cosets is surjective. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝐺)    &   𝑈 = (𝐵 / (𝐺 ~QG 𝑁))    &   𝐹 = (𝑥𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐹 = 𝑈)
 
Theoremnsgqus0 33438 A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))       ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁𝐹)
 
Theoremnsgmgclem 33439* Lemma for nsgmgc 33440. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑𝐹 ∈ (SubGrp‘𝑄))       (𝜑 → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺))
 
Theoremnsgmgc 33440* There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &   𝐽 = (𝑉MGalConn𝑊)    &   𝑉 = (toInc‘𝑆)    &   𝑊 = (toInc‘𝑇)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑𝐸𝐽𝐹)
 
Theoremnsgqusf1olem1 33441* Lemma for nsgqusf1o 33444. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
 
Theoremnsgqusf1olem2 33442* Lemma for nsgqusf1o 33444. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐸 = 𝑇)
 
Theoremnsgqusf1olem3 33443* Lemma for nsgqusf1o 33444. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐹 = 𝑆)
 
Theoremnsgqusf1o 33444* The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑𝐸:𝑆1-1-onto𝑇)
 
Theoremlmhmqusker 33445* A surjective module homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)
0 = (0g𝐻)    &   (𝜑𝐹 ∈ (𝐺 LMHom 𝐻))    &   𝐾 = (𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))       (𝜑𝐽 ∈ (𝑄 LMIso 𝐻))
 
Theoremlmicqusker 33446 The image 𝐻 of a module homomorphism 𝐹 is isomorphic with the quotient module 𝑄 over 𝐹's kernel 𝐾. This is part of what is sometimes called the first isomorphism theorem for modules. (Contributed by Thierry Arnoux, 10-Mar-2025.)
0 = (0g𝐻)    &   (𝜑𝐹 ∈ (𝐺 LMHom 𝐻))    &   𝐾 = (𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))       (𝜑𝑄𝑚 𝐻)
 
21.3.10.39  Ideals
 
Theoremlidlmcld 33447 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐼)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐼)
 
Theoremintlidl 33448 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → 𝐶 ∈ (LIdeal‘𝑅))
 
Theorem0ringidl 33449 The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{ 0 }})
 
Theorempidlnzb 33450 A principal ideal is nonzero iff it is generated by a nonzero elements (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋0 ↔ (𝐾‘{𝑋}) ≠ { 0 }))
 
Theoremlidlunitel 33451 If an ideal 𝐼 contains a unit 𝐽, then it is the whole ring. (Contributed by Thierry Arnoux, 19-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑𝐽𝑈)    &   (𝜑𝐽𝐼)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑𝐼 = 𝐵)
 
Theoremunitpidl1 33452 The ideal 𝐼 generated by an element 𝑋 of an integral domain 𝑅 is the unit ideal 𝐵 iff 𝑋 is a ring unit. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑈 = (Unit‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐼 = (𝐾‘{𝑋})    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅 ∈ IDomn)       (𝜑 → (𝐼 = 𝐵𝑋𝑈))
 
Theoremrhmquskerlem 33453* The mapping 𝐽 induced by a ring homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 22-Mar-2025.)
0 = (0g𝐻)    &   (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))    &   (𝜑𝐺 ∈ CRing)       (𝜑𝐽 ∈ (𝑄 RingHom 𝐻))
 
Theoremrhmqusker 33454* A surjective ring homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)
0 = (0g𝐻)    &   (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    &   (𝜑𝐺 ∈ CRing)    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))       (𝜑𝐽 ∈ (𝑄 RingIso 𝐻))
 
Theoremricqusker 33455 The image 𝐻 of a ring homomorphism 𝐹 is isomorphic with the quotient ring 𝑄 over 𝐹's kernel 𝐾. This a part of what is sometimes called the first isomorphism theorem for rings. (Contributed by Thierry Arnoux, 10-Mar-2025.)
0 = (0g𝐻)    &   (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    &   (𝜑𝐺 ∈ CRing)       (𝜑𝑄𝑟 𝐻)
 
Theoremelrspunidl 33456* Elementhood in the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑆 ⊆ (LIdeal‘𝑅))       (𝜑 → (𝑋 ∈ (𝑁 𝑆) ↔ ∃𝑎 ∈ (𝐵m 𝑆)(𝑎 finSupp 0𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘𝑆 (𝑎𝑘) ∈ 𝑘)))
 
Theoremelrspunsn 33457* Membership to the span of an ideal 𝑅 and a single element 𝑋. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &    + = (+g𝑅)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝑋 ∈ (𝐵𝐼))       (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
 
Theoremlidlincl 33458 Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.)
𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐽𝑈) → (𝐼𝐽) ∈ 𝑈)
 
Theoremidlinsubrg 33459 The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (LIdeal‘𝑅)    &   𝑉 = (LIdeal‘𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ∈ 𝑉)
 
Theoremrhmimaidl 33460 The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝐵 = (Base‘𝑆)    &   𝑇 = (LIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
 
Theoremdrngidl 33461 A nonzero ring is a division ring if and only if its only left ideals are the zero ideal and the unit ideal. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ NzRing → (𝑅 ∈ DivRing ↔ 𝑈 = {{ 0 }, 𝐵}))
 
Theoremdrngidlhash 33462 A ring is a division ring if and only if it admits exactly two ideals. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ Ring → (𝑅 ∈ DivRing ↔ (♯‘𝑈) = 2))
 
21.3.10.40  Prime Ideals
 
Syntaxcprmidl 33463 Extend class notation with the class of prime ideals.
class PrmIdeal
 
Definitiondf-prmidl 33464* Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵𝐼 for ideals 𝐴 and 𝐵, either 𝐴𝐼 or 𝐵𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 33469 and isprmidlc 33475. (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)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
 
Theoremprmidlval 33465* 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)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
 
Theoremisprmidl 33466* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
 
Theoremprmidlnr 33467 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃𝐵)
 
Theoremprmidl 33468* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼𝑃𝐽𝑃))
 
Theoremprmidl2 33469* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38077 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
 
Theoremidlmulssprm 33470 Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼𝑃 or 𝐽𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
× = (LSSum‘(mulGrp‘𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑃 ∈ (PrmIdeal‘𝑅))    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))    &   (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)       (𝜑 → (𝐼𝑃𝐽𝑃))
 
Theorempridln1 33471 A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ¬ 1𝐼)
 
Theoremprmidlidl 33472 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))
 
Theoremprmidlssidl 33473 Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
(𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
 
Theoremcringm4 33474 Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))
 
Theoremisprmidlc 33475* 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)    &    · = (.r𝑅)       (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))))
 
Theoremprmidlc 33476 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)    &    · = (.r𝑅)       (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼𝐵𝐽𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼𝑃𝐽𝑃))
 
Theorem0ringprmidl 33477 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) = ∅)
 
Theoremprmidl0 33478 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 = (0g𝑅)       ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)
 
Theoremrhmpreimaprmidl 33479 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)) → (𝐹𝐽) ∈ 𝑃)
 
Theoremqsidomlem1 33480 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))
 
Theoremqsidomlem2 33481 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)
 
Theoremqsidom 33482 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)))
 
Theoremqsnzr 33483 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)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑𝐼𝐵)       (𝜑𝑄 ∈ NzRing)
 
Theoremssdifidllem 33484* Lemma for ssdifidl 33485: The set 𝑃 used in the proof of ssdifidl 33485 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝑆𝐵)    &   (𝜑 → (𝑆𝐼) = ∅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆𝑝) = ∅ ∧ 𝐼𝑝)}    &   (𝜑𝑍𝑃)    &   (𝜑𝑍 ≠ ∅)    &   (𝜑 → [] Or 𝑍)       (𝜑 𝑍𝑃)
 
Theoremssdifidl 33485* 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)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝑆𝐵)    &   (𝜑 → (𝑆𝐼) = ∅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆𝑝) = ∅ ∧ 𝐼𝑝)}       (𝜑 → ∃𝑖𝑃𝑗𝑃 ¬ 𝑖𝑗)
 
Theoremssdifidlprm 33486* If the set 𝑆 of ssdifidl 33485 is multiplicatively closed, then the ideal 𝑖 is prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝑆 ∈ (SubMnd‘𝑀))    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → (𝑆𝐼) = ∅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆𝑝) = ∅ ∧ 𝐼𝑝)}       (𝜑 → ∃𝑖𝑃 (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗𝑃 ¬ 𝑖𝑗))
 
21.3.10.41  Maximal Ideals
 
Syntaxcmxidl 33487 Extend class notation with the class of maximal ideals.
class MaxIdeal
 
Definitiondf-mxidl 33488* 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)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})
 
Theoremmxidlval 33489* 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) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})
 
Theoremismxidl 33490* 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) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
 
Theoremmxidlidl 33491 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))
 
Theoremmxidlnr 33492 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)) → 𝑀𝐵)
 
Theoremmxidlmax 33493 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) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝐵))
 
Theoremmxidln1 33494 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 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1𝑀)
 
Theoremmxidlnzr 33495 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)
 
Theoremmxidlmaxv 33496 An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑅))    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝑀𝐼)    &   (𝜑𝑋 ∈ (𝐼𝑀))       (𝜑𝐼 = 𝐵)
 
Theoremcrngmxidl 33497 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 = (oppr𝑅)       (𝑅 ∈ CRing → 𝑀 = (MaxIdeal‘𝑂))
 
Theoremmxidlprm 33498 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))
 
Theoremmxidlirredi 33499 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 = (0g𝑅)    &   𝑀 = (𝐾‘{𝑋})    &   (𝜑𝑅 ∈ IDomn)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋0 )    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑅))       (𝜑𝑋 ∈ (Irred‘𝑅))
 
Theoremmxidlirred 33500 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 = (0g𝑅)    &   𝑀 = (𝐾‘{𝑋})    &   (𝜑𝑅 ∈ PID)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋0 )    &   (𝜑𝑀 ∈ (LIdeal‘𝑅))       (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))
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