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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrlmsca2 15901 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  (  _I  `  R )  =  (Scalar `  (ringLMod `  R ) )
 
Theoremrlmvsca 15902 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( .r `  R )  =  ( .s `  (ringLMod `  R )
 )
 
Theoremrlmtopn 15903 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( TopOpen `  R )  =  ( TopOpen `  (ringLMod `  R ) )
 
Theoremrlmds 15904 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( dist `  R )  =  ( dist `  (ringLMod `  R ) )
 
Theoremrlmlmod 15905 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  Ring  ->  (ringLMod `  R )  e. 
 LMod )
 
Theoremrlmlvec 15906 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e.  DivRing  ->  (ringLMod `  R )  e. 
 LVec )
 
Theoremrlmvneg 15907 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
 |-  ( inv g `  R )  =  ( inv g `  (ringLMod `  R ) )
 
Theoremrlmscaf 15908 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( + f `  (mulGrp `  R ) )  =  ( .s f `  (ringLMod `  R )
 )
 
Theoremlidlss 15909 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  B  =  ( Base `  W )   &    |-  I  =  (LIdeal `  W )   =>    |-  ( U  e.  I  ->  U  C_  B )
 
TheoremlidlssOLD 15910 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  W )   &    |-  I  =  (LIdeal `  W )   =>    |-  ( ( W  e.  V  /\  U  e.  I
 )  ->  U  C_  B )
 
Theoremislidl 15911* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( I  e.  U  <->  ( I  C_  B  /\  I  =/=  (/)  /\  A. x  e.  B  A. a  e.  I  A. b  e.  I  ( ( x 
 .x.  a )  .+  b )  e.  I
 ) )
 
Theoremlidl0cl 15912 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  .0.  e.  I
 )
 
Theoremlidlacl 15913 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .+  =  ( +g  `  R )   =>    |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  ( X  e.  I  /\  Y  e.  I )
 )  ->  ( X  .+  Y )  e.  I
 )
 
Theoremlidlnegcl 15914 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  N  =  ( inv
 g `  R )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  U  /\  X  e.  I )  ->  ( N `  X )  e.  I )
 
Theoremlidlsubg 15915 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  (LIdeal `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  I  e.  (SubGrp `  R ) )
 
Theoremlidlsubcl 15916 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  .-  =  ( -g `  R )   =>    |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  ( X  e.  I  /\  Y  e.  I )
 )  ->  ( X  .-  Y )  e.  I
 )
 
Theoremlidlmcl 15917 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 15918 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 15919 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 15920 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 15921 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 15922 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 15923 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 15924 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 15925 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 15926 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 15927 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 15928* 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 15929 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 15930* 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 15931 Ring two-sided ideal function.
 class 2Ideal
 
Definitiondf-2idl 15932 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 15933 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 15934 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 15935 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 15936 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 15937* 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 15938 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 15939 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 15940 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 15941 Ring left-principal-ideal function.
 class LPIdeal
 
Syntaxclpir 15942 Class of left principal ideal rings.
 class LPIR
 
Definitiondf-lpidl 15943* 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 15944 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 15945* 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 15946* 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 15947 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 15948 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 15949 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 15950 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 15951 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 15952 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e.  Ring )
 
Theoremdrnglpir 15953 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  ( R  e.  DivRing  ->  R  e. LPIR )
 
Theoremrspsn 15954* 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 15955* 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 15956* 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 15957 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 15958 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 15959 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 15960 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 15961 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e. NzRing  ->  R  e.  Ring )
 
Theoremdrngnzr 15962 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e.  DivRing  ->  R  e. NzRing )
 
Theoremisnzr2 15963 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 15964 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 15965 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 15966 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 15967 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 15968 Set of left-regular elements in a ring.
 class RLReg
 
Syntaxcdomn 15969 Class of (ring theoretic) domains.
 class Domn
 
Syntaxcidom 15970 Class of integral domains.
 class IDomn
 
Syntaxcpid 15971 Class of principal ideal domains.
 class PID
 
Definitiondf-rlreg 15972* 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 15973* 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 15974 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- IDomn  =  ( CRing  i^i Domn )
 
Definitiondf-pid 15975 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 15976* 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 15977* 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 15978 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 15979 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 15980 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 15981 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 15982 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 15983* 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 15984 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e. NzRing )
 
Theoremdomnrng 15985 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
 |-  ( R  e. Domn  ->  R  e.  Ring )
 
Theoremdomneq0 15986 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 15987 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 15988 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 15989 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 15990 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 15991 Another characterization of domains, hinted at in abvtriv 15554: 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 15992 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e.  DivRing  ->  R  e. Domn )
 
Theoremisidom 15993 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( R  e. IDomn  <->  ( R  e.  CRing  /\  R  e. Domn ) )
 
Theoremfldidom 15994 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( R  e. Field  ->  R  e. IDomn )
 
Theoremfidomndrnglem 15995* Lemma for fidomndrng 15996. (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 15996 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 15997 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 15998 Associative algebra.
 class AssAlg
 
Syntaxcasp 15999 Algebraic span function.
 class AlgSpan
 
Syntaxcascl 16000 Class of algebra scalar injection function.
 class algSc
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