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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrngohomcl 26701 Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  A  e.  X )  ->  ( F `
  A )  e.  Y )
 
Theoremrngohom1 26702 A ring homomorphism preserves  1. (Contributed by Jeff Madsen, 24-Jun-2011.)
 |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   &    |-  K  =  ( 2nd `  S )   &    |-  V  =  (GId `  K )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  U )  =  V )
 
Theoremrngohomadd 26703 Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   =>    |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R 
 RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A G B ) )  =  (
 ( F `  A ) J ( F `  B ) ) )
 
Theoremrngohommul 26704 Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  H  =  ( 2nd `  R )   &    |-  K  =  ( 2nd `  S )   =>    |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R 
 RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A H B ) )  =  (
 ( F `  A ) K ( F `  B ) ) )
 
Theoremrngogrphom 26705 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  J  =  ( 1st `  S )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
 
Theoremrngohom0 26706 A ring homomorphism preserves  0. (Contributed by Jeff Madsen, 2-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   &    |-  J  =  ( 1st `  S )   &    |-  W  =  (GId `  J )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  Z )  =  W )
 
Theoremrngohomsub 26707 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  H  =  (  /g  `  G )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 
 /g  `  J )   =>    |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A H B ) )  =  ( ( F `
  A ) K ( F `  B ) ) )
 
Theoremrngohomco 26708 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e. 
 RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )
 
Theoremrngokerinj 26709 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  W  =  (GId `  G )   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   &    |-  Z  =  (GId `  J )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W } )
 )
 
Definitiondf-rngoiso 26710* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  RngIso  =  ( r  e.  RingOps ,  s  e. 
 RingOps 
 |->  { f  e.  (
 r  RngHom  s )  |  f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s ) }
 )
 
Theoremrngoisoval 26711* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y } )
 
Theoremisrngoiso 26712 The predicate "is a ring isomorphism between  R and 
S." (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S ) 
 /\  F : X -1-1-onto-> Y ) ) )
 
Theoremrngoiso1o 26713 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : X
 -1-1-onto-> Y )
 
Theoremrngoisohom 26714 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
 
Theoremrngoisocnv 26715 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
 
Theoremrngoisoco 26716 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e. 
 RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngIso  T ) )
 
Definitiondf-risc 26717* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  ~=r  =  { <. r ,  s >.  |  ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) ) }
 
Theoremisriscg 26718* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  A  /\  S  e.  B ) 
 ->  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
 
Theoremisrisc 26719* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  R  e.  _V   &    |-  S  e.  _V   =>    |-  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) )
 
Theoremrisc 26720* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=r  S  <->  E. f  f  e.  ( R  RngIso  S ) ) )
 
Theoremrisci 26721 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S )
 
Theoremriscer 26722 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ~=r  Er  dom  ~=r
 
18.15.17  Commutative rings
 
Syntaxccring 26723 Extend class notation with the class of commutative rings.
 class CRingOps
 
Definitiondf-crngo 26724 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |- CRingOps  =  (
 RingOps  i^i  Com2 )
 
Theoremiscrngo 26725 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  ( R  e. CRingOps  <->  ( R  e.  RingOps  /\  R  e.  Com2 )
 )
 
Theoremiscrngo2 26726* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e. CRingOps  <->  ( R  e.  RingOps  /\ 
 A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
 
Theoremiscringd 26727* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ph  ->  G  e.  AbelOp )   &    |-  ( ph  ->  X  =  ran  G )   &    |-  ( ph  ->  H : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x H y ) H z )  =  ( x H ( y H z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  y  e.  X )  ->  (
 y H U )  =  y )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x H y )  =  ( y H x ) )   =>    |-  ( ph  ->  <. G ,  H >.  e. CRingOps )
 
Theoremcrngorngo 26728 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e. CRingOps  ->  R  e.  RingOps )
 
Theoremcrngocom 26729 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  =  ( B H A ) )
 
Theoremcrngm23 26730 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
 
Theoremcrngm4 26731 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A H B ) H ( C H D ) )  =  ( ( A H C ) H ( B H D ) ) )
 
Theoremfldcrng 26732 A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  ( K  e.  Fld  ->  K  e. CRingOps )
 
Theoremisfld2 26733 The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( K  e.  Fld  <->  ( K  e.  DivRingOps  /\  K  e. CRingOps ) )
 
Theoremcrngohomfo 26734 The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )
 
18.15.18  Ideals
 
Syntaxcidl 26735 Extend class notation with the class of ideals.
 class  Idl
 
Syntaxcpridl 26736 Extend class notation with the class of prime ideals.
 class  PrIdl
 
Syntaxcmaxidl 26737 Extend class notation with the class of maximal ideals.
 class  MaxIdl
 
Definitiondf-idl 26738* 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  e.  RingOps  |->  { i  e.  ~P ran  ( 1st `  r )  |  ( (GId `  ( 1st `  r ) )  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( 1st `  r
 ) y )  e.  i  /\  A. z  e.  ran  ( 1st `  r
 ) ( ( z ( 2nd `  r
 ) x )  e.  i  /\  ( x ( 2nd `  r
 ) z )  e.  i ) ) ) } )
 
Definitiondf-pridl 26739* Define the class of prime ideals of a ring  R. A proper ideal  I of  R is prime if whenever  A B  C_  I for ideals  A and  B, either  A  C_  I or  B  C_  I. The more familiar definition using elements rather than ideals is equivalent provided  R is commutative; see ispridl2 26766 and ispridlc 26798. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  PrIdl  =  ( r  e.  RingOps  |->  { i  e.  ( Idl `  r
 )  |  ( i  =/=  ran  ( 1st `  r )  /\  A. a  e.  ( Idl `  r ) A. b  e.  ( Idl `  r
 ) ( A. x  e.  a  A. y  e.  b  ( x ( 2nd `  r )
 y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
 
Definitiondf-maxidl 26740* 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  e.  RingOps  |->  { i  e.  ( Idl `  r
 )  |  ( i  =/=  ran  ( 1st `  r )  /\  A. j  e.  ( Idl `  r ) ( i 
 C_  j  ->  (
 j  =  i  \/  j  =  ran  ( 1st `  r ) ) ) ) } )
 
Theoremidlval 26741* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  ( Idl `  R )  =  { i  e.  ~P X  |  ( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x G y )  e.  i  /\  A. z  e.  X  ( ( z H x )  e.  i  /\  ( x H z )  e.  i ) ) ) } )
 
Theoremisidl 26742* The predicate "is an ideal of the ring  R." (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  ( I  e.  ( Idl `  R )  <->  ( I  C_  X  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  ( x G y )  e.  I  /\  A. z  e.  X  (
 ( z H x )  e.  I  /\  ( x H z )  e.  I ) ) ) ) )
 
Theoremisidlc 26743* The predicate "is an ideal of the commutative ring  R." (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e. CRingOps  ->  ( I  e.  ( Idl `  R )  <->  ( I  C_  X  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  ( x G y )  e.  I  /\  A. z  e.  X  (
 z H x )  e.  I ) ) ) )
 
Theoremidlss 26744 An ideal of  R is a subset of  R. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  I  C_  X )
 
Theoremidlcl 26745 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  X )
 
Theoremidl0cl 26746 An ideal contains  0. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  Z  e.  I )
 
Theoremidladdcl 26747 An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  I ) )  ->  ( A G B )  e.  I )
 
Theoremidllmulcl 26748 An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  X ) )  ->  ( B H A )  e.  I )
 
Theoremidlrmulcl 26749 An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  X ) )  ->  ( A H B )  e.  I )
 
Theoremidlnegcl 26750 An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  A  e.  I )  ->  ( N `
  A )  e.  I )
 
Theoremidlsubcl 26751 An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R )
 )  /\  ( A  e.  I  /\  B  e.  I ) )  ->  ( A D B )  e.  I )
 
Theoremrngoidl 26752 A ring  R is an  R ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  X  e.  ( Idl `  R )
 )
 
Theorem0idl 26753 The set containing only  0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R ) )
 
Theorem1idl 26754 Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  ( U  e.  I  <->  I  =  X ) )
 
Theorem0rngo 26755 In a ring,  0  =  1 iff the ring contains only  0. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z }
 ) )
 
Theoremdivrngidl 26756 The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  DivRingOps  ->  ( Idl `  R )  =  { { Z } ,  X } )
 
Theoremintidl 26757 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  C  =/=  (/)  /\  C  C_  ( Idl `  R ) ) 
 ->  |^| C  e.  ( Idl `  R ) )
 
Theoreminidl 26758 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R ) )  ->  ( I  i^i  J )  e.  ( Idl `  R ) )
 
Theoremunichnidl 26759* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  (
 ( R  e.  RingOps  /\  ( C  =/=  (/)  /\  C  C_  ( Idl `  R )  /\  A. i  e.  C  A. j  e.  C  ( i  C_  j  \/  j  C_  i
 ) ) )  ->  U. C  e.  ( Idl `  R ) )
 
Theoremkeridl 26760 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  S )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
 )  e.  ( Idl `  R ) )
 
Theorempridlval 26761* The class of prime ideals of a ring 
R. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( PrIdl `  R )  =  { i  e.  ( Idl `  R )  |  ( i  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R ) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  i  ->  ( a  C_  i  \/  b  C_  i ) ) ) } )
 
Theoremispridl 26762* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  ( Idl `  R ) A. b  e.  ( Idl `  R ) ( A. x  e.  a  A. y  e.  b  ( x H y )  e.  P  ->  ( a  C_  P  \/  b  C_  P ) ) ) ) )
 
Theorempridlidl 26763 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R ) )  ->  P  e.  ( Idl `  R ) )
 
Theorempridlnr 26764 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R ) )  ->  P  =/=  X )
 
Theorempridl 26765* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  H  =  ( 2nd `  R )   =>    |-  ( ( ( R  e.  RingOps  /\  P  e.  ( PrIdl `  R )
 )  /\  ( A  e.  ( Idl `  R )  /\  B  e.  ( Idl `  R )  /\  A. x  e.  A  A. y  e.  B  ( x H y )  e.  P ) )  ->  ( A  C_  P  \/  B  C_  P ) )
 
Theoremispridl2 26766* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 26798 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P ) ) ) )  ->  P  e.  ( PrIdl `  R )
 )
 
Theoremmaxidlval 26767* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( MaxIdl `  R )  =  {
 i  e.  ( Idl `  R )  |  ( i  =/=  X  /\  A. j  e.  ( Idl `  R ) ( i 
 C_  j  ->  (
 j  =  i  \/  j  =  X ) ) ) } )
 
Theoremismaxidl 26768* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( M  e.  ( MaxIdl `  R ) 
 <->  ( M  e.  ( Idl `  R )  /\  M  =/=  X  /\  A. j  e.  ( Idl `  R ) ( M 
 C_  j  ->  (
 j  =  M  \/  j  =  X )
 ) ) ) )
 
Theoremmaxidlidl 26769 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  (
 ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  M  e.  ( Idl `  R ) )
 
Theoremmaxidlnr 26770 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  M  =/=  X )
 
Theoremmaxidlmax 26771 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  /\  ( I  e.  ( Idl `  R )  /\  M  C_  I ) ) 
 ->  ( I  =  M  \/  I  =  X ) )
 
Theoremmaxidln1 26772 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  -.  U  e.  M )
 
Theoremmaxidln0 26773 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  M  e.  ( MaxIdl `  R ) )  ->  U  =/=  Z )
 
18.15.19  Prime rings and integral domains
 
Syntaxcprrng 26774 Extend class notation with the class of prime rings.
 class  PrRing
 
Syntaxcdmn 26775 Extend class notation with the class of domains.
 class  Dmn
 
Definitiondf-prrngo 26776 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  e.  RingOps  |  {
 (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r ) }
 
Definitiondf-dmn 26777 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  Dmn  =  ( PrRing  i^i  Com2 )
 
Theoremisprrngo 26778 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  PrRing  <->  ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) ) )
 
Theoremprrngorngo 26779 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  PrRing  ->  R  e. 
 RingOps )
 
Theoremsmprngopr 26780 A simple ring (one whose only ideals are  0 and  R) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  U  =/=  Z  /\  ( Idl `  R )  =  { { Z } ,  X } )  ->  R  e.  PrRing )
 
Theoremdivrngpr 26781 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  DivRingOps  ->  R  e.  PrRing )
 
Theoremisdmn 26782 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 )
 )
 
Theoremisdmn2 26783 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
 
Theoremdmncrng 26784 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  Dmn  ->  R  e. CRingOps )
 
Theoremdmnrngo 26785 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  Dmn  ->  R  e. 
 RingOps )
 
Theoremflddmn 26786 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( K  e.  Fld  ->  K  e.  Dmn )
 
18.15.20  Ideal generators
 
Syntaxcigen 26787 Extend class notation with the ideal generation function.
 class  IdlGen
 
Definitiondf-igen 26788* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^| { j  e.  ( Idl `  r
 )  |  s  C_  j } )
 
Theoremigenval 26789* 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  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
 
Theoremigenss 26790 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
 
Theoremigenidl 26791 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R ) )
 
Theoremigenmin 26792 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R )  /\  S  C_  I )  ->  ( R  IdlGen  S )  C_  I )
 
Theoremigenidl2 26793 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  ( R  IdlGen  I )  =  I )
 
Theoremigenval2 26794* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( ( R  IdlGen  S )  =  I  <->  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I 
 C_  j ) ) ) )
 
Theoremprnc 26795* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  A  e.  X )  ->  ( R  IdlGen  { A } )  =  { x  e.  X  |  E. y  e.  X  x  =  ( y H A ) } )
 
Theoremisfldidl 26796 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  K )   &    |-  H  =  ( 2nd `  K )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( K  e.  Fld  <->  ( K  e. CRingOps  /\  U  =/=  Z 
 /\  ( Idl `  K )  =  { { Z } ,  X }
 ) )
 
Theoremisfldidl2 26797 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  K )   &    |-  H  =  ( 2nd `  K )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( K  e.  Fld  <->  ( K  e. CRingOps  /\  X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
 
Theoremispridlc 26798* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e. CRingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  (
 ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P )
 ) ) ) )
 
Theorempridlc 26799 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P )
 )  ->  ( A  e.  P  \/  B  e.  P ) )
 
Theorempridlc2 26800 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  ( X  \  P ) 
 /\  B  e.  X  /\  ( A H B )  e.  P )
 )  ->  B  e.  P )
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