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Type | Label | Description |
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Statement | ||
Syntax | crisc 32201 | Extend class notation with the ring isomorphism relation. |
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Definition | df-rngohom 32202* | Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | rngohomval 32203* | The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.) |
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Theorem | isrngohom 32204* |
The predicate "is a ring homomorphism from ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohomf 32205 | A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohomcl 32206 | Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohom1 32207 |
A ring homomorphism preserves ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohomadd 32208 | Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohommul 32209 | Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.) |
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Theorem | rngogrphom 32210 | A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohom0 32211 |
A ring homomorphism preserves ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohomsub 32212 | Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngohomco 32213 | The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | rngokerinj 32214 | A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Definition | df-rngoiso 32215* | Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | rngoisoval 32216* | The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isrngoiso 32217 |
The predicate "is a ring isomorphism between ![]() ![]() |
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Theorem | rngoiso1o 32218 | A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngoisohom 32219 | A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | rngoisocnv 32220 | The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rngoisoco 32221 | The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-risc 32222* | Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | isriscg 32223* | The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | isrisc 32224* | The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | risc 32225* | The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | risci 32226 | Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | riscer 32227 | Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.) |
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Syntax | ccring 32228 | Extend class notation with the class of commutative rings. |
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Definition | df-crngo 32229 | Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.) |
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Theorem | iscrngo 32230 | The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
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Theorem | iscrngo2 32231* | The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | iscringd 32232* | Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | crngorngo 32233 | A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | crngocom 32234 | The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | crngm23 32235 | Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | crngm4 32236 | Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fldcrng 32237 | A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) |
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Theorem | isfld2 32238 | The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | crngohomfo 32239 | The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | cidl 32240 | Extend class notation with the class of ideals. |
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Syntax | cpridl 32241 | Extend class notation with the class of prime ideals. |
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Syntax | cmaxidl 32242 | Extend class notation with the class of maximal ideals. |
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Definition | df-idl 32243* |
Define the class of (two-sided) ideals of a ring ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-pridl 32244* |
Define the class of prime ideals of a ring ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-maxidl 32245* |
Define the class of maximal ideals of a ring ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | idlval 32246* | The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isidl 32247* |
The predicate "is an ideal of the ring ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isidlc 32248* |
The predicate "is an ideal of the commutative ring ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | idlss 32249 |
An ideal of ![]() ![]() |
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Theorem | idlcl 32250 | An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | idl0cl 32251 |
An ideal contains ![]() |
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Theorem | idladdcl 32252 | An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | idllmulcl 32253 | An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | idlrmulcl 32254 | An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | idlnegcl 32255 | An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | idlsubcl 32256 | An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | rngoidl 32257 |
A ring ![]() ![]() |
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Theorem | 0idl 32258 |
The set containing only ![]() |
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Theorem | 1idl 32259 | Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | 0rngo 32260 |
In a ring, ![]() ![]() ![]() ![]() |
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Theorem | divrngidl 32261 | The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | intidl 32262 | The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | inidl 32263 | The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | unichnidl 32264* | The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
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Theorem | keridl 32265 | The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.) |
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Theorem | pridlval 32266* |
The class of prime ideals of a ring ![]() |
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Theorem | ispridl 32267* | The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | pridlidl 32268 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | pridlnr 32269 | A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | pridl 32270* | The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | ispridl2 32271* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 32303 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) |
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Theorem | maxidlval 32272* | The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) |
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Theorem | ismaxidl 32273* | The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) |
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Theorem | maxidlidl 32274 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
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Theorem | maxidlnr 32275 | A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | maxidlmax 32276 | A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | maxidln1 32277 | One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) |
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Theorem | maxidln0 32278 | A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
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Syntax | cprrng 32279 | Extend class notation with the class of prime rings. |
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Syntax | cdmn 32280 | Extend class notation with the class of domains. |
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Definition | df-prrngo 32281 | 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.) |
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Definition | df-dmn 32282 | Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | isprrngo 32283 | The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | prrngorngo 32284 | A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | smprngopr 32285 |
A simple ring (one whose only ideals are ![]() ![]() |
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Theorem | divrngpr 32286 | A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
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Theorem | isdmn 32287 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | isdmn2 32288 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | dmncrng 32289 | A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
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Theorem | dmnrngo 32290 | A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
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Theorem | flddmn 32291 | A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Syntax | cigen 32292 | Extend class notation with the ideal generation function. |
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Definition | df-igen 32293* | Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | igenval 32294* | The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.) |
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Theorem | igenss 32295 | A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | igenidl 32296 | The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | igenmin 32297 | The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | igenidl2 32298 | The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | igenval2 32299* | The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | prnc 32300* | A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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