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Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrngohom0 33901 A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)    &   𝐽 = (1st𝑆)    &   𝑊 = (GId‘𝐽)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑍) = 𝑊)

Theoremrngohomsub 33902 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐻 = ( /𝑔𝐺)    &   𝐽 = (1st𝑆)    &   𝐾 = ( /𝑔𝐽)       (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))

Theoremrngohomco 33903 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
(((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngHom 𝑇))

Theoremrngokerinj 33904 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑊 = (GId‘𝐺)    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽    &   𝑍 = (GId‘𝐽)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))

Definitiondf-rngoiso 33905* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})

Theoremrngoisoval 33906* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})

Theoremisrngoiso 33907 The predicate "is a ring isomorphism between 𝑅 and 𝑆." (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋1-1-onto𝑌)))

Theoremrngoiso1o 33908 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝑋1-1-onto𝑌)

Theoremrngoisohom 33909 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))

Theoremrngoisocnv 33910 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))

Theoremrngoisoco 33911 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
(((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngIso 𝑇))

Definitiondf-risc 33912* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}

Theoremisriscg 33913* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))

Theoremisrisc 33914* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑅 ∈ V    &   𝑆 ∈ V       (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))

Theoremrisc 33915* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))

Theoremrisci 33916 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅𝑟 𝑆)

Theoremriscer 33917 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝑟 Er dom ≃𝑟

20.19.19  Commutative rings

Syntaxccm2 33918 Extend class notation with a class that adds commutativity to various flavors of rings.
class Com2

Definitiondf-com2 33919* A device to add commutativity to various sorts of rings. I use ran 𝑔 because I suppose 𝑔 has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Com2 = {⟨𝑔, ⟩ ∣ ∀𝑎 ∈ ran 𝑔𝑏 ∈ ran 𝑔(𝑎𝑏) = (𝑏𝑎)}

Syntaxcfld 33920 Extend class notation with the class of all fields.
class Fld

Definitiondf-fld 33921 Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
Fld = (DivRingOps ∩ Com2)

Syntaxccring 33922 Extend class notation with the class of commutative rings.
class CRingOps

Definitiondf-crngo 33923 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps = (RingOps ∩ Com2)

Theoremiscom2 33924* A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))

Theoremiscrngo 33925 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
(𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Theoremiscrngo2 33926* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))

Theoremiscringd 33927* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
(𝜑𝐺 ∈ AbelOp)    &   (𝜑𝑋 = ran 𝐺)    &   (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))    &   (𝜑𝑈𝑋)    &   ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))       (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps)

Theoremflddivrng 33928 A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
(𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Theoremcrngorngo 33929 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Theoremcrngocom 33930 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

Theoremcrngm23 33931 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))

Theoremcrngm4 33932 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Theoremfldcrng 33933 A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
(𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Theoremisfld2 33934 The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))

Theoremcrngohomfo 33935 The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ CRingOps)

20.19.20  Ideals

Syntaxcidl 33936 Extend class notation with the class of ideals.
class Idl

Syntaxcpridl 33937 Extend class notation with the class of prime ideals.
class PrIdl

Syntaxcmaxidl 33938 Extend class notation with the class of maximal ideals.
class MaxIdl

Definitiondf-idl 33939* Define the class of (two-sided) ideals of a ring 𝑅. A subset of 𝑅 is an ideal if it contains 0, is closed under addition, and is closed under multiplication on either side by any element of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st𝑟) ∣ ((GId‘(1st𝑟)) ∈ 𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖)))})

Definitiondf-pridl 33940* 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 ispridl2 33967 and ispridlc 33999. (Contributed by Jeff Madsen, 10-Jun-2010.)
PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Definitiondf-maxidl 33941* 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.)
MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))))})

Theoremidlval 33942* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})

Theoremisidl 33943* The predicate "is an ideal of the ring 𝑅." (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))

Theoremisidlc 33944* The predicate "is an ideal of the commutative ring 𝑅." (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))

Theoremidlss 33945 An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)

Theoremidlcl 33946 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴𝑋)

Theoremidl0cl 33947 An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍𝐼)

𝐺 = (1st𝑅)       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)

Theoremidllmulcl 33949 An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)

Theoremidlrmulcl 33950 An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)

Theoremidlnegcl 33951 An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑁 = (inv‘𝐺)       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)

Theoremidlsubcl 33952 An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐷 = ( /𝑔𝐺)       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)

Theoremrngoidl 33953 A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))

Theorem0idl 33954 The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Theorem1idl 33955 Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))

Theorem0rngo 33956 In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))

Theoremdivrngidl 33957 The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Theoremintidl 33958 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))

Theoreminidl 33959 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼𝐽) ∈ (Idl‘𝑅))

Theoremunichnidl 33960* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ∈ (Idl‘𝑅))

Theoremkeridl 33961 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
𝐺 = (1st𝑆)    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Theorempridlval 33962* The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Theoremispridl 33963* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))

Theorempridlidl 33964 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))

Theorempridlnr 33965 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)

Theorempridl 33966* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐻 = (2nd𝑅)       (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))

Theoremispridl2 33967* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 33999 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))

Theoremmaxidlval 33968* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})

Theoremismaxidl 33969* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))

Theoremmaxidlidl 33970 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))

Theoremmaxidlnr 33971 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀𝑋)

Theoremmaxidlmax 33972 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝑋))

Theoremmaxidln1 33973 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐻 = (2nd𝑅)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)

Theoremmaxidln0 33974 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)

20.19.21  Prime rings and integral domains

Syntaxcprrng 33975 Extend class notation with the class of prime rings.
class PrRing

Syntaxcdmn 33976 Extend class notation with the class of domains.
class Dmn

Definitiondf-prrngo 33977 Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}

Definitiondf-dmn 33978 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Dmn = (PrRing ∩ Com2)

Theoremisprrngo 33979 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))

Theoremprrngorngo 33980 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Theoremsmprngopr 33981 A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Theoremdivrngpr 33982 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Theoremisdmn 33983 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Theoremisdmn2 33984 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))

Theoremdmncrng 33985 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Theoremdmnrngo 33986 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Theoremflddmn 33987 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld → 𝐾 ∈ Dmn)

20.19.22  Ideal generators

Syntaxcigen 33988 Extend class notation with the ideal generation function.
class IdlGen

Definitiondf-igen 33989* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})

Theoremigenval 33990* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})

Theoremigenss 33991 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Theoremigenidl 33992 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))

Theoremigenmin 33993 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)

Theoremigenidl2 33994 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Theoremigenval2 33995* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))

Theoremprnc 33996* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})

Theoremisfldidl 33997 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝐾)    &   𝐻 = (2nd𝐾)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Theoremisfldidl2 33998 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝐾)    &   𝐻 = (2nd𝐾)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Theoremispridlc 33999* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))

Theorempridlc 34000 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))

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