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Theorem List for Metamath Proof Explorer - 33001-33100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprmidl2 33001* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 37405 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 33002 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 33003 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 33004 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 33005 Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
(𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
 
Theoremlidlnsg 33006 An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
 
Theoremcringm4 33007 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 33008* 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 33009 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 33010 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 33011 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 33012 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 33013 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 33014 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 33015 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 33016 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)
 
21.3.9.31  Maximal Ideals
 
Syntaxcmxidl 33017 Extend class notation with the class of maximal ideals.
class MaxIdeal
 
Definitiondf-mxidl 33018* 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 33019* 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 33020* 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 33021 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 33022 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 33023 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 33024 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 33025 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 33026 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 33027 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 33028 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 33029 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 33030 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)))
 
Theoremssmxidllem 33031* The set 𝑃 used in the proof of ssmxidl 33032 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐼𝐵)    &   (𝜑𝑍𝑃)    &   (𝜑𝑍 ≠ ∅)    &   (𝜑 → [] Or 𝑍)       (𝜑 𝑍𝑃)
 
Theoremssmxidl 33032* Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)
 
Theoremdrnglidl1ne0 33033 In a nonzero ring, the zero ideal is different of the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing → 𝐵 ≠ { 0 })
 
Theoremdrng0mxidl 33034 In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
0 = (0g𝑅)       (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅))
 
Theoremdrngmxidl 33035 The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.)
0 = (0g𝑅)       (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }})
 
Theoremkrull 33036* Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))
 
Theoremmxidlnzrb 33037* A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)))
 
Theoremopprabs 33038 The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑 → Fun 𝑅)    &   (𝜑 → (.r‘ndx) ∈ dom 𝑅)    &   (𝜑· Fn (𝐵 × 𝐵))       (𝜑𝑅 = (oppr𝑂))
 
Theoremoppreqg 33039 Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅𝑉𝐼𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
 
Theoremopprnsg 33040 Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑂 = (oppr𝑅)       (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
 
Theoremopprlidlabs 33041 The ideals of the opposite ring's opposite ring are the ideals of the original ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr𝑂)))
 
Theoremoppr2idl 33042 Two sided ideal of the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
 
Theoremopprmxidlabs 33043 The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑅))       (𝜑𝑀 ∈ (MaxIdeal‘(oppr𝑂)))
 
Theoremopprqusbas 33044 The base of the quotient of the opposite ring is the same as the base of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝐵)       (𝜑 → (Base‘(oppr𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
 
Theoremopprqusplusg 33045 The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐸 = (Base‘𝑄)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)       (𝜑 → (𝑋(+g‘(oppr𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
 
Theoremopprqus0g 33046 The group identity element of the quotient of the opposite ring is the same as the group identity element of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑𝐼 ∈ (NrmSGrp‘𝑅))       (𝜑 → (0g‘(oppr𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
 
Theoremopprqusmulr 33047 The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (2Ideal‘𝑅))    &   𝐸 = (Base‘𝑄)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)       (𝜑 → (𝑋(.r‘(oppr𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
 
Theoremopprqus1r 33048 The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (2Ideal‘𝑅))       (𝜑 → (1r‘(oppr𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
 
Theoremopprqusdrng 33049 The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (2Ideal‘𝑅))       (𝜑 → ((oppr𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))
 
Theoremqsdrngilem 33050* Lemma for qsdrngi 33051. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑅))    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑂))    &   (𝜑𝑋 ∈ (Base‘𝑅))    &   (𝜑 → ¬ 𝑋𝑀)       (𝜑 → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r𝑄))
 
Theoremqsdrngi 33051 A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑅))    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑂))       (𝜑𝑄 ∈ DivRing)
 
Theoremqsdrnglem2 33052 Lemma for qsdrng 33053. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑀 ∈ (2Ideal‘𝑅))    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑄 ∈ DivRing)    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))    &   (𝜑𝑀𝐽)    &   (𝜑𝑋 ∈ (𝐽𝑀))       (𝜑𝐽 = 𝐵)
 
Theoremqsdrng 33053 An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑂 = (oppr𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑀 ∈ (2Ideal‘𝑅))       (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))
 
Theoremqsfld 33054 An ideal 𝑀 in the commutative ring 𝑅 is maximal if and only if the factor ring 𝑄 is a field. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑀 ∈ (LIdeal‘𝑅))       (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
 
TheoremmxidlprmALT 33055 Every maximal ideal is prime - alternative proof. (Contributed by Thierry Arnoux, 15-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑅 ∈ CRing)    &   (𝜑𝑀 ∈ (MaxIdeal‘𝑅))       (𝜑𝑀 ∈ (PrmIdeal‘𝑅))
 
21.3.9.32  The semiring of ideals of a ring
 
Syntaxcidlsrg 33056 Extend class notation with the semiring of ideals of a ring.
class IDLsrg
 
Definitiondf-idlsrg 33057* Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
IDLsrg = (𝑟 ∈ V ↦ (LIdeal‘𝑟) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩}))
 
Theoremidlsrgstr 33058 A constructed semiring of ideals is a structure. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩})       𝑊 Struct ⟨1, 10⟩
 
Theoremidlsrgval 33059* Lemma for idlsrgbas 33060 through idlsrgtset 33064. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐼 = (LIdeal‘𝑅)    &    = (LSSum‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (LSSum‘𝐺)       (𝑅𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))
 
Theoremidlsrgbas 33060 Base of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐼 = (LIdeal‘𝑅)       (𝑅𝑉𝐼 = (Base‘𝑆))
 
Theoremidlsrgplusg 33061 Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &    = (LSSum‘𝑅)       (𝑅𝑉 = (+g𝑆))
 
Theoremidlsrg0g 33062 The zero ideal is the additive identity of the semiring of ideals. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } = (0g𝑆))
 
Theoremidlsrgmulr 33063* Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (LSSum‘𝐺)       (𝑅𝑉 → (𝑖𝐵, 𝑗𝐵 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗))) = (.r𝑆))
 
Theoremidlsrgtset 33064* Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐽 = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})       (𝑅𝑉𝐽 = (TopSet‘𝑆))
 
Theoremidlsrgmulrval 33065 Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &   𝐺 = (mulGrp‘𝑅)    &    · = (LSSum‘𝐺)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽)))
 
Theoremidlsrgmulrcl 33066 Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ∈ 𝐵)
 
Theoremidlsrgmulrss1 33067 In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ⊆ 𝐼)
 
Theoremidlsrgmulrss2 33068 The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ⊆ 𝐽)
 
Theoremidlsrgmulrssin 33069 In a commutative ring, the product of two ideals is a subset of their intersection. (Contributed by Thierry Arnoux, 17-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ⊆ (𝐼𝐽))
 
Theoremidlsrgmnd 33070 The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Mnd)
 
Theoremidlsrgcmnd 33071 The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ CMnd)
 
21.3.9.33  Unique factorization domains
 
Syntaxcufd 33072 Class of unique factorization domains.
class UFD
 
Definitiondf-ufd 33073* Define the class of unique factorization domains. A unique factorization domain (UFD for short), is a commutative ring with an absolute value (from abvtriv 20680 this is equivalent to being a domain) such that every prime ideal contains a prime element (this is a characterization due to Irving Kaplansky). A UFD is sometimes also called a "factorial ring" following the terminology of Bourbaki. (Contributed by Mario Carneiro, 17-Feb-2015.)
UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)}
 
Theoremisufd 33074* The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐴 = (AbsVal‘𝑅)    &   𝐼 = (PrmIdeal‘𝑅)    &   𝑃 = (RPrime‘𝑅)       (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖𝐼 (𝑖𝑃) ≠ ∅)))
 
Theoremrprmval 33075* The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (∥r𝑅)       (𝑅𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥𝐵𝑦𝐵 (𝑝 (𝑥 · 𝑦) → (𝑝 𝑥𝑝 𝑦))})
 
Theoremisrprm 33076* Property for 𝑃 to be a prime element in the ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)       (𝑅𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥𝐵𝑦𝐵 (𝑃 (𝑥 · 𝑦) → (𝑃 𝑥𝑃 𝑦)))))
 
21.3.9.34  Associative algebras
 
Theoremasclmulg 33077 Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (.g𝑊)    &    = (.g𝐹)       ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾) → (𝐴‘(𝑁 𝑋)) = (𝑁 (𝐴𝑋)))
 
21.3.9.35  Univariate Polynomials
 
Theorem0ringmon1p 33078 There are no monic polynomials over a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
𝑀 = (Monic1p𝑅)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑 → (♯‘𝐵) = 1)       (𝜑𝑀 = ∅)
 
Theoremfply1 33079 Conditions for a function to be a univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.)
0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑃 = (Base‘(Poly1𝑅))    &   (𝜑𝐹:(ℕ0m 1o)⟶𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑𝐹𝑃)
 
Theoremply1lvec 33080 In a division ring, the univariate polynomials form a vector space. (Contributed by Thierry Arnoux, 19-Feb-2025.)
𝑃 = (Poly1𝑅)    &   (𝜑𝑅 ∈ DivRing)       (𝜑𝑃 ∈ LVec)
 
Theoremply1scleq 33081 Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → ((𝐴𝐸) = (𝐴𝐹) ↔ 𝐸 = 𝐹))
 
Theoremevls1fn 33082 Functionality of the subring polynomial evaluation. (Contributed by Thierry Arnoux, 9-Feb-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑃 = (Poly1‘(𝑅s 𝑆))    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))       (𝜑𝑂 Fn 𝑈)
 
Theoremevls1dm 33083 The domain of the subring polynomial evaluation function. (Contributed by Thierry Arnoux, 9-Feb-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑃 = (Poly1‘(𝑅s 𝑆))    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))       (𝜑 → dom 𝑂 = 𝑈)
 
Theoremevls1fvf 33084 The subring evaluation function for a univariate polynomial as a function, with domain and codomain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑃 = (Poly1‘(𝑅s 𝑆))    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑄𝑈)       (𝜑 → (𝑂𝑄):𝐵𝐵)
 
Theoremevls1scafv 33085 Value of the univariate polynomial evaluation for scalars. (Contributed by Thierry Arnoux, 21-Jan-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)    &   (𝜑𝐶𝐵)       (𝜑 → ((𝑄‘(𝐴𝑋))‘𝐶) = 𝑋)
 
Theoremevls1expd 33086 Univariate polynomial evaluation builder for an exponential. See also evl1expd 22185. (Contributed by Thierry Arnoux, 24-Jan-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &    = (.g‘(mulGrp‘𝑊))    &    = (.g‘(mulGrp‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑀𝐵)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑁 𝑀))‘𝐶) = (𝑁 ((𝑄𝑀)‘𝐶)))
 
Theoremevls1varpwval 33087 Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 22202. (Contributed by Thierry Arnoux, 24-Jan-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝑋 = (var1𝑈)    &   𝐵 = (Base‘𝑆)    &    = (.g‘(mulGrp‘𝑊))    &    = (.g‘(mulGrp‘𝑆))    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐵)       (𝜑 → ((𝑄‘(𝑁 𝑋))‘𝐶) = (𝑁 𝐶))
 
Theoremevls1fpws 33088* Evaluation of a univariate subring polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑀𝐵)    &    · = (.r𝑆)    &    = (.g‘(mulGrp‘𝑆))    &   𝐴 = (coe1𝑀)       (𝜑 → (𝑄𝑀) = (𝑥𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑥))))))
 
Theoremressply1evl 33089 Evaluation of a univariate subring polynomial is the same as the evaluation in the bigger ring. (Contributed by Thierry Arnoux, 23-Jan-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑊)    &   𝐸 = (eval1𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑𝑄 = (𝐸𝐵))
 
Theoremevls1addd 33090 Univariate polynomial evaluation of a sum of polynomials. (Contributed by Thierry Arnoux, 8-Feb-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑊)    &    = (+g𝑊)    &    + = (+g𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑀𝐵)    &   (𝜑𝑁𝐵)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑀 𝑁))‘𝐶) = (((𝑄𝑀)‘𝐶) + ((𝑄𝑁)‘𝐶)))
 
Theoremevls1muld 33091 Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑊)    &    × = (.r𝑊)    &    · = (.r𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑀𝐵)    &   (𝜑𝑁𝐵)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄𝑀)‘𝐶) · ((𝑄𝑁)‘𝐶)))
 
Theoremevls1vsca 33092 Univariate polynomial evaluation of a scalar product of polynomials. (Contributed by Thierry Arnoux, 25-Feb-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑊)    &    × = ( ·𝑠𝑊)    &    · = (.r𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐴𝑅)    &   (𝜑𝑁𝐵)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝐴 × 𝑁))‘𝐶) = (𝐴 · ((𝑄𝑁)‘𝐶)))
 
Theoremressdeg1 33093 The degree of a univariate polynomial in a structure restriction. (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝐻 = (𝑅s 𝑇)    &   𝐷 = ( deg1𝑅)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑃𝐵)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))       (𝜑 → (𝐷𝑃) = (( deg1𝐻)‘𝑃))
 
Theoremply1ascl0 33094 The zero scalar as a polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝑊 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝑂 = (0g𝑅)    &    0 = (0g𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝐴𝑂) = 0 )
 
Theoremply1ascl1 33095 The multiplicative unit scalar as a univariate polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝑊 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝐼 = (1r𝑅)    &    1 = (1r𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝐴𝐼) = 1 )
 
Theoremply1asclunit 33096 A non-zero scalar polynomial is a unit. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Field)    &   (𝜑𝑌𝐵)    &   (𝜑𝑌0 )       (𝜑 → (𝐴𝑌) ∈ (Unit‘𝑃))
 
Theoremdeg1le0eq0 33097 A polynomial with nonpositive degree is the zero polynomial iff its constant term is zero. Biconditional version of deg1scl 25970. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑂 = (0g𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑 → (𝐷𝐹) ≤ 0)       (𝜑 → (𝐹 = 𝑂 ↔ ((coe1𝐹)‘0) = 0 ))
 
Theoremressply10g 33098 A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑍 = (0g𝑆)       (𝜑𝑍 = (0g𝑈))
 
Theoremressply1mon1p 33099 The monic polynomials of a restricted polynomial algebra. (Contributed by Thierry Arnoux, 21-Jan-2025.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑀 = (Monic1p𝑅)    &   𝑁 = (Monic1p𝐻)       (𝜑𝑁 = (𝐵𝑀))
 
Theoremressply1invg 33100 An element of a restricted polynomial algebra has the same group inverse. (Contributed by Thierry Arnoux, 30-Jan-2025.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → ((invg𝑈)‘𝑋) = ((invg𝑃)‘𝑋))
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