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Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlidlmcl 15801 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  ( X  e.  B  /\  Y  e.  I )
 )  ->  ( X  .x.  Y )  e.  I
 )
 
Theoremlidl1el 15802 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  (  .1.  e.  I 
 <->  I  =  B ) )
 
Theoremlidl0 15803 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
 
Theoremlidl1 15804 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  B  e.  U )
 
Theoremlidlacs 15805 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  W )   &    |-  I  =  (LIdeal `  W )   =>    |-  ( W  e.  Ring  ->  I  e.  (ACS `  B ) )
 
Theoremrspcl 15806 The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  Ring  /\  G  C_  B )  ->  ( K `  G )  e.  U )
 
Theoremrspssid 15807 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  G  C_  B )  ->  G  C_  ( K `  G ) )
 
Theoremrsp1 15808 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( K `  {  .1.  } )  =  B )
 
Theoremrsp0 15809 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( K `  {  .0.  } )  =  {  .0.  } )
 
Theoremrspssp 15810 The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U  /\  G  C_  I )  ->  ( K `  G )  C_  I )
 
Theoremmrcrsp 15811 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  F  =  (mrCls `  U )   =>    |-  ( R  e.  Ring  ->  K  =  F )
 
Theoremlidlnz 15812* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U  /\  I  =/=  {  .0.  } )  ->  E. x  e.  I  x  =/=  .0.  )
 
Theoremdrngnidl 15813 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
 
Theoremlidlrsppropd 15814* The left ideals and ring span of a ring depend only on the ring components. Here  W is expected to be either 
B (when closure is available) or  _V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  C_  W )   &    |-  ( ( ph  /\  ( x  e.  W  /\  y  e.  W ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( .r `  K ) y )  e.  W )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r
 `  L ) y ) )   =>    |-  ( ph  ->  (
 (LIdeal `  K )  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L )
 ) )
 
10.8.2  Two-sided ideals and quotient rings
 
Syntaxc2idl 15815 Ring two-sided ideal function.
 class 2Ideal
 
Definitiondf-2idl 15816 Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
 |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr `  r ) ) ) )
 
Theorem2idlval 15817 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  I  =  (LIdeal `  R )   &    |-  O  =  (oppr `  R )   &    |-  J  =  (LIdeal `  O )   &    |-  T  =  (2Ideal `  R )   =>    |-  T  =  ( I  i^i  J )
 
Theorem2idlcpbl 15818 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  R )   &    |-  E  =  ( R ~QG 
 S )   &    |-  I  =  (2Ideal `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( ( A E C  /\  B E D )  ->  ( A  .x.  B ) E ( C  .x.  D ) ) )
 
Theoremdivs1 15819 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (2Ideal `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  S  e.  I ) 
 ->  ( U  e.  Ring  /\ 
 [  .1.  ] ( R ~QG  S )  =  ( 1r
 `  U ) ) )
 
Theoremdivsrng 15820 If  S is a two-sided ideal in  R, then  U  =  R  /  S is a ring, called the quotient ring of 
R by  S. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (2Ideal `  R )   =>    |-  (
 ( R  e.  Ring  /\  S  e.  I ) 
 ->  U  e.  Ring )
 
Theoremdivsrhm 15821* If  S is a two-sided ideal in  R, then the "natural map" from elements to their cosets is a ring homomorphism from  R to  R  /  S. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (2Ideal `  R )   &    |-  X  =  ( Base `  R )   &    |-  F  =  ( x  e.  X  |->  [ x ] ( R ~QG  S ) )   =>    |-  ( ( R  e.  Ring  /\  S  e.  I ) 
 ->  F  e.  ( R RingHom  U ) )
 
Theoremcrngridl 15822 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  I  =  (LIdeal `  R )   &    |-  O  =  (oppr `  R )   =>    |-  ( R  e.  CRing  ->  I  =  (LIdeal `  O ) )
 
Theoremcrng2idl 15823 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  I  =  (LIdeal `  R )   =>    |-  ( R  e.  CRing  ->  I  =  (2Ideal `  R ) )
 
Theoremdivscrng 15824 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  CRing  /\  S  e.  I ) 
 ->  U  e.  CRing )
 
10.8.3  Principal ideal rings. Divisibility in the integers
 
Syntaxclpidl 15825 Ring left-principal-ideal function.
 class LPIdeal
 
Syntaxclpir 15826 Class of left principal ideal rings.
 class LPIR
 
Definitiondf-lpidl 15827* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |- LPIdeal  =  ( w  e.  Ring  |->  U_ g  e.  ( Base `  w ) { (
 (RSpan `  w ) `  { g } ) } )
 
Definitiondf-lpir 15828 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |- LPIR  =  { w  e.  Ring  |  (LIdeal `  w )  =  (LPIdeal `  w ) }
 
Theoremlpival 15829* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  Ring 
 ->  P  =  U_ g  e.  B  { ( K `
  { g }
 ) } )
 
Theoremislpidl 15830* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  Ring 
 ->  ( I  e.  P  <->  E. g  e.  B  I  =  ( K `  { g } ) ) )
 
Theoremlpi0 15831 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  {  .0.  }  e.  P )
 
Theoremlpi1 15832 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  B  e.  P )
 
Theoremislpir 15833 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LPIR  <->  ( R  e.  Ring  /\  U  =  P ) )
 
Theoremlpiss 15834 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e.  Ring  ->  P  C_  U )
 
Theoremislpir2 15835 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LPIR  <->  ( R  e.  Ring  /\  U  C_  P )
 )
 
Theoremlpirrng 15836 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e.  Ring )
 
Theoremdrnglpir 15837 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  ( R  e.  DivRing  ->  R  e. LPIR )
 
Theoremrspsn 15838* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  K  =  (RSpan `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  G  e.  B ) 
 ->  ( K `  { G } )  =  { x  |  G  .||  x }
 )
 
Theoremlidldvgen 15839* An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (LIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U  /\  G  e.  B ) 
 ->  ( I  =  ( K `  { G } )  <->  ( G  e.  I  /\  A. x  e.  I  G  .||  x ) ) )
 
Theoremlpigen 15840* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  P  =  (LPIdeal `  R )   &    |-  .||  =  ( ||r
 `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  ( I  e.  P  <->  E. x  e.  I  A. y  e.  I  x  .||  y ) )
 
10.8.4  Nonzero rings
 
Syntaxcnzr 15841 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 15842 A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- NzRing  =  { r  e.  Ring  |  ( 1r `  r
 )  =/=  ( 0g `  r ) }
 
Theoremisnzr 15843 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. NzRing  <->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
 )
 
Theoremnzrnz 15844 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
 
Theoremnzrrng 15845 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e. NzRing  ->  R  e.  Ring )
 
Theoremdrngnzr 15846 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e.  DivRing  ->  R  e. NzRing )
 
Theoremisnzr2 15847 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e. NzRing  <->  ( R  e.  Ring  /\  2o  ~<_  B ) )
 
Theoremopprnzr 15848 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. NzRing  ->  O  e. NzRing )
 
Theoremrngelnzr 15849 A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  ( B  \  {  .0.  } ) )  ->  R  e. NzRing )
 
Theoremnzrunit 15850 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. NzRing  /\  A  e.  U ) 
 ->  A  =/=  .0.  )
 
Theoremsubrgnzr 15851 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  S  =  ( Rs  A )   =>    |-  ( ( R  e. NzRing  /\  A  e.  (SubRing `  R ) )  ->  S  e. NzRing )
 
10.8.5  Left regular elements. More kinds of ring
 
Syntaxcrlreg 15852 Set of left-regular elements in a ring.
 class RLReg
 
Syntaxcdomn 15853 Class of (ring theoretic) domains.
 class Domn
 
Syntaxcidom 15854 Class of integral domains.
 class IDomn
 
Syntaxcpid 15855 Class of principal ideal domains.
 class PID
 
Definitiondf-rlreg 15856* Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |- RLReg  =  ( r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r
 ) ) } )
 
Definitiondf-domn 15857* A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |- Domn  =  { r  e. NzRing  |  [. ( Base `  r )  /  b ]. [. ( 0g `  r )  /  z ]. A. x  e.  b  A. y  e.  b  ( ( x ( .r `  r
 ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) ) }
 
Definitiondf-idom 15858 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- IDomn  =  ( CRing  i^i Domn )
 
Definitiondf-pid 15859 A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |- PID 
 =  (IDomn  i^i LPIR )
 
Theoremrrgval 15860* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x 
 .x.  y )  =  .0.  ->  y  =  .0.  ) }
 
Theoremisrrg 15861* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y
 )  =  .0.  ->  y  =  .0.  ) ) )
 
Theoremrrgeq0i 15862 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X 
 .x.  Y )  =  .0. 
 ->  Y  =  .0.  )
 )
 
Theoremrrgeq0 15863 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0.  <->  Y  =  .0.  ) )
 
Theoremrrgsupp 15864 Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  E )   &    |-  ( ph  ->  Y : I --> B )   =>    |-  ( ph  ->  ( `' ( ( I  X.  { X } )  o F  .x.  Y ) " ( _V  \  {  .0.  } ) )  =  ( `' Y "
 ( _V  \  {  .0.  } ) ) )
 
Theoremrrgss 15865 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  E  C_  B
 
Theoremunitrrg 15866 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  E  =  (RLReg `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e.  Ring  ->  U  C_  E )
 
Theoremisdomn 15867* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. Domn  <->  ( R  e. NzRing  /\ 
 A. x  e.  B  A. y  e.  B  ( ( x  .x.  y
 )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
 ) ) )
 
Theoremdomnnzr 15868 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e. NzRing )
 
Theoremdomnrng 15869 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e.  Ring )
 
Theoremdomneq0 15870 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
 
Theoremdomnmuln0 15871 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/=  .0.  )
 )  ->  ( X  .x.  Y )  =/=  .0.  )
 
Theoremisdomn2 15872 A ring is a domain iff all nonzero elements are non-zero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  E  =  (RLReg `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. Domn  <->  ( R  e. NzRing  /\  ( B  \  {  .0.  } )  C_  E ) )
 
Theoremdomnrrg 15873 In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  E  =  (RLReg `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/=  .0.  )  ->  X  e.  E )
 
Theoremopprdomn 15874 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. Domn  ->  O  e. Domn )
 
Theoremabvn0b 15875 Another characterization of domains, hinted at in abvtriv 15441: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  A  =  (AbsVal `  R )   =>    |-  ( R  e. Domn  <->  ( R  e. NzRing  /\  A  =/=  (/) ) )
 
Theoremdrngdomn 15876 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e.  DivRing  ->  R  e. Domn )
 
Theoremisidom 15877 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( R  e. IDomn  <->  ( R  e.  CRing  /\  R  e. Domn ) )
 
Theoremfldidom 15878 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e. Field  ->  R  e. IDomn )
 
Theoremfidomndrnglem 15879* Lemma for fidomndrng 15880. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e. Domn )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A  e.  ( B  \  {  .0.  }
 ) )   &    |-  F  =  ( x  e.  B  |->  ( x  .x.  A )
 )   =>    |-  ( ph  ->  A  .|| 
 .1.  )
 
Theoremfidomndrng 15880 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( B  e.  Fin  ->  ( R  e. Domn  <->  R  e.  DivRing ) )
 
Theoremfiidomfld 15881 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( B  e.  Fin  ->  ( R  e. IDomn  <->  R  e. Field ) )
 
10.9  Associative algebras
 
10.9.1  Definition and basic properties
 
Syntaxcasa 15882 Associative algebra.
 class AssAlg
 
Syntaxcasp 15883 Algebraic span function.
 class AlgSpan
 
Syntaxcascl 15884 Class of algebra scalar injection function.
 class algSc
 
Definitiondf-assa 15885* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with an multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |- AssAlg  =  { w  e.  ( LMod  i^i  Ring )  |  [. (Scalar `  w )  /  f ]. ( f  e. 
 CRing  /\  A. r  e.  ( Base `  f ) A. x  e.  ( Base `  w ) A. y  e.  ( Base `  w ) [. ( .s `  w )  /  s ]. [. ( .r
 `  w )  /  t ]. ( ( ( r s x ) t y )  =  ( r s ( x t y ) )  /\  ( x t ( r s y ) )  =  ( r s ( x t y ) ) ) ) }
 
Definitiondf-asp 15886* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w )  i^i  ( LSubSp `  w ) )  |  s  C_  t } ) )
 
Definitiondf-ascl 15887* Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unit. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
 |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w ) )  |->  ( x ( .s `  w ) ( 1r `  w ) ) ) )
 
Theoremisassa 15888* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( W  e. AssAlg  <->  ( ( W  e.  LMod  /\  W  e.  Ring  /\  F  e.  CRing )  /\  A. r  e.  B  A. x  e.  V  A. y  e.  V  ( ( ( r  .x.  x )  .X.  y )  =  ( r  .x.  ( x  .X.  y ) )  /\  ( x  .X.  ( r 
 .x.  y ) )  =  ( r  .x.  ( x  .X.  y ) ) ) ) )
 
Theoremassalem 15889 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( (
 ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) ) 
 /\  ( X  .X.  ( A  .x.  Y ) )  =  ( A 
 .x.  ( X  .X.  Y ) ) ) )
 
Theoremassaass 15890 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( ( A  .x.  X )  .X.  Y )  =  ( A 
 .x.  ( X  .X.  Y ) ) )
 
Theoremassaassr 15891 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  (
 Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  W )   =>    |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( X  .X.  ( A  .x.  Y ) )  =  ( A  .x.  ( X  .X.  Y ) ) )
 
Theoremassalmod 15892 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( W  e. AssAlg  ->  W  e.  LMod )
 
Theoremassarng 15893 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( W  e. AssAlg  ->  W  e.  Ring )
 
Theoremassasca 15894 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. AssAlg  ->  F  e.  CRing )
 
Theoremisassad 15895* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( ph  ->  V  =  ( Base `  W )
 )   &    |-  ( ph  ->  F  =  (Scalar `  W )
 )   &    |-  ( ph  ->  B  =  ( Base `  F )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  .X. 
 =  ( .r `  W ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  W  e.  Ring
 )   &    |-  ( ph  ->  F  e.  CRing )   &    |-  ( ( ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  ->  ( ( r  .x.  x )  .X.  y )  =  ( r  .x.  ( x  .X.  y ) ) )   &    |-  ( ( ph  /\  (
 r  e.  B  /\  x  e.  V  /\  y  e.  V )
 )  ->  ( x  .X.  ( r  .x.  y
 ) )  =  ( r  .x.  ( x  .X.  y ) ) )   =>    |-  ( ph  ->  W  e. AssAlg )
 
Theoremissubassa 15896 The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( Ws  A )   &    |-  L  =  (
 LSubSp `  W )   &    |-  V  =  ( Base `  W )   &    |-  .1.  =  ( 1r `  W )   =>    |-  ( ( W  e. AssAlg  /\ 
 .1.  e.  A  /\  A  C_  V )  ->  ( S  e. AssAlg  <->  ( A  e.  (SubRing `  W )  /\  A  e.  L )
 ) )
 
Theoremsraassa 15897 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   =>    |-  ( ( W  e.  CRing  /\  S  e.  (SubRing `  W ) )  ->  A  e. AssAlg )
 
Theoremrlmassa 15898 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( R  e.  CRing  ->  (ringLMod `  R )  e. AssAlg )
 
Theoremassapropd 15899* If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   &    |-  ( ph  ->  F  =  (Scalar `  K ) )   &    |-  ( ph  ->  F  =  (Scalar `  L ) )   &    |-  P  =  (
 Base `  F )   &    |-  (
 ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s `  L ) y ) )   =>    |-  ( ph  ->  ( K  e. AssAlg  <->  L  e. AssAlg ) )
 
Theoremaspval 15900* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W )  i^i  L )  |  S  C_  t } )
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