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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-lidl 16201 Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |- LIdeal  =  ( LSubSp  o. ringLMod )
 
Definitiondf-rsp 16202 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |- RSpan  =  ( LSpan  o. ringLMod )
 
Theoremsraval 16203 Lemma for srabase 16205 through sravsca 16209. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |-  ( ( W  e.  V  /\  S  C_  ( Base `  W ) ) 
 ->  ( ( subringAlg  `  W ) `
  S )  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s
 `  ndx ) ,  ( .r `  W ) >. ) )
 
Theoremsralem 16204 Lemma for srabase 16205 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  ( N  <  5  \/  6  <  N )   =>    |-  ( ph  ->  ( E `  W )  =  ( E `  A ) )
 
Theoremsrabase 16205 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( Base `  W )  =  (
 Base `  A ) )
 
Theoremsraaddg 16206 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( +g  `  W )  =  (
 +g  `  A )
 )
 
Theoremsramulr 16207 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( .r `  W )  =  ( .r `  A ) )
 
Theoremsrasca 16208 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )
 
Theoremsravsca 16209 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( .r `  W )  =  ( .s `  A ) )
 
Theoremsratset 16210 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  (TopSet `  W )  =  (TopSet `  A ) )
 
Theoremsratopn 16211 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( TopOpen `  W )  =  ( TopOpen `  A ) )
 
Theoremsrads 16212 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( dist `  W )  =  (
 dist `  A ) )
 
Theoremsralmod 16213 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   =>    |-  ( S  e.  (SubRing `  W )  ->  A  e.  LMod )
 
Theoremsralmod0 16214 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  W ) )   &    |-  ( ph  ->  S 
 C_  ( Base `  W ) )   =>    |-  ( ph  ->  .0.  =  ( 0g `  A ) )
 
Theoremissubgrpd2 16215* Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  S  =  ( Is  D ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  I ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  I )
 )   &    |-  ( ph  ->  D  C_  ( Base `  I )
 )   &    |-  ( ph  ->  .0.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .+  y
 )  e.  D )   &    |-  ( ( ph  /\  x  e.  D )  ->  (
 ( inv g `  I
 ) `  x )  e.  D )   &    |-  ( ph  ->  I  e.  Grp )   =>    |-  ( ph  ->  D  e.  (SubGrp `  I
 ) )
 
Theoremissubgrpd 16216* Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  S  =  ( Is  D ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  I ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  I )
 )   &    |-  ( ph  ->  D  C_  ( Base `  I )
 )   &    |-  ( ph  ->  .0.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .+  y
 )  e.  D )   &    |-  ( ( ph  /\  x  e.  D )  ->  (
 ( inv g `  I
 ) `  x )  e.  D )   &    |-  ( ph  ->  I  e.  Grp )   =>    |-  ( ph  ->  S  e.  Grp )
 
Theoremissubrngd2 16217* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  S  =  ( Is  D ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  I ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  I )
 )   &    |-  ( ph  ->  D  C_  ( Base `  I )
 )   &    |-  ( ph  ->  .0.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .+  y
 )  e.  D )   &    |-  ( ( ph  /\  x  e.  D )  ->  (
 ( inv g `  I
 ) `  x )  e.  D )   &    |-  ( ph  ->  .1. 
 =  ( 1r `  I ) )   &    |-  ( ph  ->  .x.  =  ( .r `  I ) )   &    |-  ( ph  ->  .1.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .x.  y
 )  e.  D )   &    |-  ( ph  ->  I  e.  Ring
 )   =>    |-  ( ph  ->  D  e.  (SubRing `  I )
 )
 
Theoremrlmfn 16218 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  Fn  _V
 
Theoremrlmval 16219 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `
  ( Base `  W ) )
 
Theoremlidlval 16220 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (LIdeal `  W )  =  ( LSubSp `  (ringLMod `  W ) )
 
Theoremrspval 16221 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (RSpan `  W )  =  ( LSpan `  (ringLMod `  W ) )
 
Theoremrlmbas 16222 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( Base `  R )  =  ( Base `  (ringLMod `  R ) )
 
Theoremrlmplusg 16223 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R )
 )
 
Theoremrlm0 16224 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( 0g `  R )  =  ( 0g `  (ringLMod `  R )
 )
 
Theoremrlmmulr 16225 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( .r `  R )  =  ( .r `  (ringLMod `  R )
 )
 
Theoremrlmsca 16226 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  X  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
 
Theoremrlmsca2 16227 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  (  _I  `  R )  =  (Scalar `  (ringLMod `  R ) )
 
Theoremrlmvsca 16228 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( .r `  R )  =  ( .s `  (ringLMod `  R )
 )
 
Theoremrlmtopn 16229 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( TopOpen `  R )  =  ( TopOpen `  (ringLMod `  R ) )
 
Theoremrlmds 16230 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( dist `  R )  =  ( dist `  (ringLMod `  R ) )
 
Theoremrlmlmod 16231 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  Ring  ->  (ringLMod `  R )  e. 
 LMod )
 
Theoremrlmlvec 16232 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 16233 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 16234 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( + f `  (mulGrp `  R ) )  =  ( .s f `  (ringLMod `  R )
 )
 
Theoremlidlss 16235 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 16236 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 16237* 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 16238 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 16239 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 16240 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 16241 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 16242 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 16243 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 16244 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 16245 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 16246 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 16247 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 16248 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 16249 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 16250 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 16251 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 16252 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 16253 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 16254* 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 16255 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 16256* 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 16257 Ring two-sided ideal function.
 class 2Ideal
 
Definitiondf-2idl 16258 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 16259 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 16260 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 16261 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 16262 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 16263* 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 16264 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 16265 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 16266 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 16267 Ring left-principal-ideal function.
 class LPIdeal
 
Syntaxclpir 16268 Class of left principal ideal rings.
 class LPIR
 
Definitiondf-lpidl 16269* 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 16270 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 16271* 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 16272* 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 16273 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 16274 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 16275 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 16276 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 16277 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 16278 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e.  Ring )
 
Theoremdrnglpir 16279 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  ( R  e.  DivRing  ->  R  e. LPIR )
 
Theoremrspsn 16280* 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 16281* 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 16282* 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 16283 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 16284 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 16285 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 16286 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 16287 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e. NzRing  ->  R  e.  Ring )
 
Theoremdrngnzr 16288 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( R  e.  DivRing  ->  R  e. NzRing )
 
Theoremisnzr2 16289 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 16290 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 16291 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 16292 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 16293 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 rings
 
Syntaxcrlreg 16294 Set of left-regular elements in a ring.
 class RLReg
 
Syntaxcdomn 16295 Class of (ring theoretic) domains.
 class Domn
 
Syntaxcidom 16296 Class of integral domains.
 class IDomn
 
Syntaxcpid 16297 Class of principal ideal domains.
 class PID
 
Definitiondf-rlreg 16298* 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 16299* 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 16300 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- IDomn  =  ( CRing  i^i Domn )
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