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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrngogrphom 26601 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
GrpOpHom

Theoremrngohom0 26602 A ring homomorphism preserves . (Contributed by Jeff Madsen, 2-Jan-2011.)
GId              GId

Theoremrngohomsub 26603 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)

Theoremrngohomco 26604 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngokerinj 26605 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
GId                     GId

Definitiondf-rngoiso 26606* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisoval 26607* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisrngoiso 26608 The predicate "is a ring isomorphism between and ." (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoiso1o 26609 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisohom 26610 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisocnv 26611 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrngoisoco 26612 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Definitiondf-risc 26613* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisriscg 26614* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremisrisc 26615* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrisc 26616* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremrisci 26617 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremriscer 26618 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)

19.14.16  Commutative rings

Syntaxccring 26619 Extend class notation with the class of commutative rings.
CRingOps

Definitiondf-crngo 26620 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremiscrngo 26621 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremiscrngo2 26622* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremiscringd 26623* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
CRingOps

Theoremcrngorngo 26624 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremcrngocom 26625 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremcrngm23 26626 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

Theoremcrngm4 26627 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

Theoremfldcrng 26628 A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps

Theoremisfld2 26629 The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremcrngohomfo 26630 The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
CRingOps CRingOps

19.14.17  Ideals

Syntaxcidl 26631 Extend class notation with the class of ideals.

Syntaxcpridl 26632 Extend class notation with the class of prime ideals.

Syntaxcmaxidl 26633 Extend class notation with the class of maximal ideals.

Definitiondf-idl 26634* Define the class of (two-sided) ideals of a ring . A subset of is an ideal if it contains , is closed under addition, and is closed under multiplication on either side by any element of . (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Definitiondf-pridl 26635* Define the class of prime ideals of a ring . A proper ideal of is prime if whenever for ideals and , either or . The more familiar definition using elements rather than ideals is equivalent provided is commutative; see ispridl2 26662 and ispridlc 26694. (Contributed by Jeff Madsen, 10-Jun-2010.)

Definitiondf-maxidl 26636* Define the class of maximal ideals of a ring . A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremidlval 26637* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremisidl 26638* The predicate "is an ideal of the ring ." (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremisidlc 26639* The predicate "is an ideal of the commutative ring ." (Contributed by Jeff Madsen, 10-Jun-2010.)
GId       CRingOps

Theoremidlss 26640 An ideal of is a subset of . (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlcl 26641 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremidl0cl 26642 An ideal contains . (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremidllmulcl 26644 An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlrmulcl 26645 An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlnegcl 26646 An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremidlsubcl 26647 An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremrngoidl 26648 A ring is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theorem0idl 26649 The set containing only is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theorem1idl 26650 Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theorem0rngo 26651 In a ring, iff the ring contains only . (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       GId

Theoremdivrngidl 26652 The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremintidl 26653 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoreminidl 26654 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremunichnidl 26655* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremkeridl 26656 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
GId

Theorempridlval 26657* The class of prime ideals of a ring . (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremispridl 26658* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)

Theorempridlidl 26659 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theorempridlnr 26660 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theorempridl 26661* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremispridl2 26662* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 26694 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)

Theoremmaxidlval 26663* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremismaxidl 26664* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremmaxidlidl 26665 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremmaxidlnr 26666 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremmaxidlmax 26667 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremmaxidln1 26668 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
GId

Theoremmaxidln0 26669 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
GId       GId

19.14.18  Prime rings and integral domains

Syntaxcprrng 26670 Extend class notation with the class of prime rings.

Syntaxcdmn 26671 Extend class notation with the class of domains.

Definitiondf-prrngo 26672 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.)
GId

Definitiondf-dmn 26673 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremisprrngo 26674 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
GId

Theoremprrngorngo 26675 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremsmprngopr 26676 A simple ring (one whose only ideals are and ) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       GId

Theoremdivrngpr 26677 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremisdmn 26678 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremisdmn2 26679 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremdmncrng 26680 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
CRingOps

Theoremdmnrngo 26681 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremflddmn 26682 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)

19.14.19  Ideal generators

Syntaxcigen 26683 Extend class notation with the ideal generation function.

Definitiondf-igen 26684* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenval 26685* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)

Theoremigenss 26686 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenidl 26687 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenmin 26688 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenidl2 26689 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremigenval2 26690* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremprnc 26691* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
CRingOps

Theoremisfldidl 26692 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
GId       GId       CRingOps

Theoremisfldidl2 26693 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId       CRingOps

Theoremispridlc 26694* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
CRingOps

Theorempridlc 26695 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theorempridlc2 26696 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theorempridlc3 26697 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
CRingOps

Theoremisdmn3 26698* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
GId       GId       CRingOps

Theoremdmnnzd 26699 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
GId

Theoremdmncan1 26700 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
GId

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