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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremintop 38901 An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( M intOp  N )  ->  .o.  : ( M  X.  M ) --> N )
 
Theoremclintopval 38902 The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  ( clIntOp  `  M )  =  ( M  ^m  ( M  X.  M ) ) )
 
Theoremassintopval 38903* The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  ( assIntOp  `  M )  =  {
 o  e.  ( clIntOp  `  M )  |  o assLaw  M }
 )
 
Theoremassintopmap 38904* The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  ( assIntOp  `  M )  =  {
 o  e.  ( M 
 ^m  ( M  X.  M ) )  |  o assLaw  M } )
 
Theoremisclintop 38905 The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  (  .o.  e.  ( clIntOp  `  M ) 
 <->  .o.  : ( M  X.  M ) --> M ) )
 
Theoremclintop 38906 A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( clIntOp  `  M ) 
 ->  .o.  : ( M  X.  M ) --> M )
 
Theoremassintop 38907 An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  (  .o.  : ( M  X.  M ) --> M  /\  .o. assLaw  M ) )
 
Theoremisassintop 38908* The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  (  .o.  e.  ( assIntOp  `  M ) 
 <->  (  .o.  : ( M  X.  M ) --> M  /\  A. x  e.  M  A. y  e.  M  A. z  e.  M  ( ( x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) ) )
 
Theoremclintopcllaw 38909 The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( clIntOp  `  M ) 
 ->  .o. clLaw  M )
 
Theoremassintopcllaw 38910 The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  .o. clLaw  M )
 
Theoremassintopasslaw 38911 The associative low holds for a associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  .o. assLaw  M )
 
Theoremassintopass 38912* An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  A. x  e.  M  A. y  e.  M  A. z  e.  M  (
 ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) )
 
21.33.12.3  Alternative definitions for Magmas and Semigroups
 
Syntaxcmgm2 38913 Extend class notation with class of all magmas.
 class MgmALT
 
Syntaxccmgm2 38914 Extend class notation with class of all commutative magmas.
 class CMgmALT
 
Syntaxcsgrp2 38915 Extend class notation with class of all semigroups.
 class SGrpALT
 
Syntaxccsgrp2 38916 Extend class notation with class of all commutative semigroups.
 class CSGrpALT
 
Definitiondf-mgm2 38917 A magma is a set equipped with a closed operation. Definition 1 of [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by AV, 6-Jan-2020.)
 |- MgmALT  =  { m  |  ( +g  `  m ) clLaw  ( Base `  m ) }
 
Definitiondf-cmgm2 38918 A commutative magma is a magma with a commutative operation. Definition 8 of [BourbakiAlg1] p. 7. (Contributed by AV, 20-Jan-2020.)
 |- CMgmALT  =  { m  e. MgmALT  |  ( +g  `  m ) comLaw  ( Base `  m ) }
 
Definitiondf-sgrp2 38919 A semigroup is a magma with an associative operation. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4, or of a semi-group in section 1.3 of [Hall] p. 7. (Contributed by AV, 6-Jan-2020.)
 |- SGrpALT  =  {
 g  e. MgmALT  |  ( +g  `  g ) assLaw  ( Base `  g ) }
 
Definitiondf-csgrp2 38920 A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
 |- CSGrpALT  =  {
 g  e. SGrpALT  |  ( +g  `  g ) comLaw  ( Base `  g ) }
 
TheoremismgmALT 38921 The predicate "is a magma." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e.  V  ->  ( M  e. MgmALT  <->  .o. clLaw  B ) )
 
TheoremiscmgmALT 38922 The predicate "is a commutative magma." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e. CMgmALT  <->  ( M  e. MgmALT  /\ 
 .o. comLaw  B ) )
 
TheoremissgrpALT 38923 The predicate "is a semigroup." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e. SGrpALT  <->  ( M  e. MgmALT  /\ 
 .o. assLaw  B ) )
 
TheoremiscsgrpALT 38924 The predicate "is a commutative semigroup." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e. CSGrpALT  <->  ( M  e. SGrpALT  /\ 
 .o. comLaw  B ) )
 
Theoremmgm2mgm 38925 Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
 |-  ( M  e. MgmALT  <->  M  e. Mgm )
 
Theoremsgrp2sgrp 38926 Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
 |-  ( M  e. SGrpALT  <->  M  e. SGrp )
 
21.33.13  Categories (extension)
 
21.33.13.1  Subcategories (extension)
 
Theoremidfusubc0 38927* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
 |-  S  =  ( C  |`cat  J )   &    |-  I  =  (idfunc `  S )   &    |-  B  =  (
 Base `  S )   =>    |-  ( J  e.  (Subcat `  C )  ->  I  =  <. (  _I  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x ( Hom  `  S ) y ) ) ) >. )
 
Theoremidfusubc 38928* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
 |-  S  =  ( C  |`cat  J )   &    |-  I  =  (idfunc `  S )   &    |-  B  =  (
 Base `  S )   =>    |-  ( J  e.  (Subcat `  C )  ->  I  =  <. (  _I  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x J y ) ) ) >. )
 
Theoreminclfusubc 38929* The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.)
 |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  S  =  ( C  |`cat  J )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  F  =  (  _I  |`  B )
 )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x J y ) ) ) )   =>    |-  ( ph  ->  F ( S  Func  C ) G )
 
21.33.14  Rings (extension)
 
21.33.14.1  Nonzero rings (extension)
 
Theoremlmod0rng 38930 If the scalar ring of a module is the zero ring, the module is the zero module, i.e. the base set of the module is the singleton consisting of the identity element only. (Contributed by AV, 17-Apr-2019.)
 |-  (
 ( M  e.  LMod  /\ 
 -.  (Scalar `  M )  e. NzRing )  ->  ( Base `  M )  =  {
 ( 0g `  M ) } )
 
Theoremnzrneg1ne0 38931 The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.)
 |-  ( R  e. NzRing  ->  ( ( invg `  R ) `  ( 1r `  R ) )  =/=  ( 0g `  R ) )
 
Theorem0ringdif 38932 A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  ( Ring  \ NzRing )  <->  ( R  e.  Ring  /\  B  =  {  .0.  } ) )
 
Theorem0ringbas 38933 The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  ( Ring  \ NzRing )  ->  B  =  {  .0.  } )
 
Theorem0ring1eq0 38934 In a zero ring, a ring which is not a nonzero ring, the unit equals the zero element. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  ( Ring  \ NzRing )  ->  .1.  =  .0.  )
 
Theoremnrhmzr 38935 There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)
 |-  (
 ( Z  e.  ( Ring  \ NzRing )  /\  R  e. NzRing )  ->  ( Z RingHom  R )  =  (/) )
 
21.33.14.2  Non-unital rings ("rngs")

According to Wikipedia, "... in abstract algebra, a rng (or pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 6-Jan-2020).

 
Syntaxcrng 38936 Extend class notation with class of all non-unital rings.
 class Rng
 
Definitiondf-rng0 38937* Define class of all (non-unital) rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
 |- Rng  =  {
 f  e.  Abel  |  ( (mulGrp `  f )  e. SGrp  /\  [. ( Base `  f
 )  /  b ]. [. ( +g  `  f
 )  /  p ]. [. ( .r `  f )  /  t ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( ( x t ( y p z ) )  =  ( ( x t y ) p ( x t z ) )  /\  ( ( x p y ) t z )  =  ( ( x t z ) p ( y t z ) ) ) ) }
 
Theoremisrng 38938* The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
 |-  B  =  ( Base `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e. Rng  <->  ( R  e.  Abel  /\  G  e. SGrp  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( ( x 
 .x.  ( y  .+  z ) )  =  ( ( x  .x.  y )  .+  ( x 
 .x.  z ) ) 
 /\  ( ( x 
 .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) ) ) )
 
Theoremrngabl 38939 A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
 |-  ( R  e. Rng  ->  R  e.  Abel
 )
 
Theoremrngmgp 38940 A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e. Rng  ->  G  e. SGrp )
 
Theoremringrng 38941 A unital ring is a (non-unital) ring. (Contributed by AV, 6-Jan-2020.)
 |-  ( R  e.  Ring  ->  R  e. Rng )
 
Theoremringssrng 38942 The unital rings are (non-unital) rings. (Contributed by AV, 20-Mar-2020.)
 |-  Ring  C_ Rng
 
Theoremisringrng 38943* The predicate "is a unital ring" as extension of the predicate "is a non-unital ring". (Contributed by AV, 17-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e.  Ring  <->  ( R  e. Rng  /\ 
 E. x  e.  B  A. y  e.  B  ( ( x  .x.  y
 )  =  y  /\  ( y  .x.  x )  =  y ) ) )
 
Theoremrngdir 38944 Distributive law for the multiplication operation of a nonunital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. Rng  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )
 
Theoremrngcl 38945 Closure of the multiplication operation of a nonunital ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. Rng  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
 
Theoremrnglz 38946 The zero of a nonunital ring is a left-absorbing element. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Rng  /\  X  e.  B ) 
 ->  (  .0.  .x.  X )  =  .0.  )
 
21.33.14.3  Rng homomorphisms
 
Syntaxcrngh 38947 non-unital ring homomorphisms.
 class RngHomo
 
Syntaxcrngs 38948 non-unital ring isomorphisms.
 class RngIsom
 
Definitiondf-rnghomo 38949* Define the set of non-unital ring homomorphisms from  r to  s. (Contributed by AV, 20-Feb-2020.)
 |- RngHomo  =  ( r  e. Rng ,  s  e. Rng 
 |->  [_ ( Base `  r
 )  /  v ]_ [_ ( Base `  s )  /  w ]_ { f  e.  ( w  ^m  v
 )  |  A. x  e.  v  A. y  e.  v  ( ( f `
  ( x (
 +g  `  r )
 y ) )  =  ( ( f `  x ) ( +g  `  s ) ( f `
  y ) ) 
 /\  ( f `  ( x ( .r `  r ) y ) )  =  ( ( f `  x ) ( .r `  s
 ) ( f `  y ) ) ) } )
 
Definitiondf-rngisom 38950* Define the set of non-unital ring isomorphisms from  r to  s. (Contributed by AV, 20-Feb-2020.)
 |- RngIsom  =  ( r  e.  _V ,  s  e.  _V  |->  { f  e.  ( r RngHomo  s )  |  `' f  e.  ( s RngHomo  r ) } )
 
Theoremrnghmrcl 38951 Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020.)
 |-  ( F  e.  ( R RngHomo  S )  ->  ( R  e. Rng  /\  S  e. Rng )
 )
 
Theoremrnghmfn 38952 The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
 |- RngHomo  Fn  (Rng  X. Rng
 )
 
Theoremrnghmval 38953* The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .*  =  ( .r `  S )   &    |-  C  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   =>    |-  (
 ( R  e. Rng  /\  S  e. Rng )  ->  ( R RngHomo  S )  =  {
 f  e.  ( C 
 ^m  B )  | 
 A. x  e.  B  A. y  e.  B  ( ( f `  ( x  .+  y ) )  =  ( ( f `
  x )  .+b  ( f `  y
 ) )  /\  (
 f `  ( x  .x.  y ) )  =  ( ( f `  x )  .*  (
 f `  y )
 ) ) } )
 
Theoremisrnghm 38954* A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .*  =  ( .r `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  <->  ( ( R  e. Rng  /\  S  e. Rng ) 
 /\  ( F  e.  ( R  GrpHom  S ) 
 /\  A. x  e.  B  A. y  e.  B  ( F `  ( x 
 .x.  y ) )  =  ( ( F `
  x )  .*  ( F `  y ) ) ) ) )
 
Theoremisrnghmmul 38955 A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
 |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  <->  ( ( R  e. Rng  /\  S  e. Rng ) 
 /\  ( F  e.  ( R  GrpHom  S ) 
 /\  F  e.  ( M MgmHom  N ) ) ) )
 
Theoremrnghmmgmhm 38956 A non-unital ring homomorphism is a homomorphism of multiplicative magmas. (Contributed by AV, 27-Feb-2020.)
 |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  ->  F  e.  ( M MgmHom  N ) )
 
Theoremrnghmval2 38957 The non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 1-Mar-2020.)
 |-  (
 ( R  e. Rng  /\  S  e. Rng )  ->  ( R RngHomo  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MgmHom  (mulGrp `  S )
 ) ) )
 
Theoremisrngisom 38958 An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
 |-  (
 ( R  e.  V  /\  S  e.  W ) 
 ->  ( F  e.  ( R RngIsom  S )  <->  ( F  e.  ( R RngHomo  S )  /\  `' F  e.  ( S RngHomo  R ) ) ) )
 
Theoremrngimrcl 38959 Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
 |-  ( F  e.  ( R RngIsom  S )  ->  ( R  e.  _V  /\  S  e.  _V ) )
 
Theoremrnghmghm 38960 A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  ( F  e.  ( R RngHomo  S )  ->  F  e.  ( R  GrpHom  S ) )
 
Theoremrnghmf 38961 A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  ->  F : B
 --> C )
 
Theoremrnghmmul 38962 A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.)
 |-  X  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  S )   =>    |-  ( ( F  e.  ( R RngHomo  S )  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  ( A  .x.  B ) )  =  ( ( F `
  A )  .X.  ( F `  B ) ) )
 
Theoremisrnghm2d 38963* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  S )   &    |-  ( ph  ->  R  e. Rng )   &    |-  ( ph  ->  S  e. Rng )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( F `
  ( x  .x.  y ) )  =  ( ( F `  x )  .X.  ( F `
  y ) ) )   &    |-  ( ph  ->  F  e.  ( R  GrpHom  S ) )   =>    |-  ( ph  ->  F  e.  ( R RngHomo  S )
 )
 
Theoremisrnghmd 38964* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  S )   &    |-  ( ph  ->  R  e. Rng )   &    |-  ( ph  ->  S  e. Rng )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( F `
  ( x  .x.  y ) )  =  ( ( F `  x )  .X.  ( F `
  y ) ) )   &    |-  C  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+^  =  (
 +g  `  S )   &    |-  ( ph  ->  F : B --> C )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( F `
  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
  y ) ) )   =>    |-  ( ph  ->  F  e.  ( R RngHomo  S )
 )
 
Theoremrnghmf1o 38965 A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  ->  ( F : B -1-1-onto-> C  <->  `' F  e.  ( S RngHomo  R ) ) )
 
Theoremisrngim 38966 An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  (
 ( R  e.  V  /\  S  e.  W ) 
 ->  ( F  e.  ( R RngIsom  S )  <->  ( F  e.  ( R RngHomo  S )  /\  F : B -1-1-onto-> C ) ) )
 
Theoremrngimf1o 38967 An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngIsom  S )  ->  F : B
 -1-1-onto-> C )
 
Theoremrngimrnghm 38968 An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngIsom  S )  ->  F  e.  ( R RngHomo  S ) )
 
Theoremrnghmco 38969 The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( T RngHomo  U )  /\  G  e.  ( S RngHomo  T )
 )  ->  ( F  o.  G )  e.  ( S RngHomo  U ) )
 
Theoremidrnghm 38970 The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e. Rng  ->  (  _I  |`  B )  e.  ( R RngHomo  R ) )
 
Theoremc0mgm 38971* The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e. Mgm  /\  T  e.  Mnd )  ->  H  e.  ( S MgmHom  T ) )
 
Theoremc0mhm 38972* The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Mnd  /\  T  e.  Mnd )  ->  H  e.  ( S MndHom  T ) )
 
Theoremc0ghm 38973* The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Grp  /\  T  e.  Grp )  ->  H  e.  ( S 
 GrpHom  T ) )
 
Theoremc0rhm 38974* The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Ring  /\  T  e.  ( Ring  \ NzRing
 ) )  ->  H  e.  ( S RingHom  T )
 )
 
Theoremc0rnghm 38975* The constant mapping to zero is a nonunital ring homomorphism from any nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
 ) )  ->  H  e.  ( S RngHomo  T )
 )
 
Theoremc0snmgmhm 38976* The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Mnd  /\  T  e. Mgm  /\  ( # `
  B )  =  1 )  ->  H  e.  ( T MgmHom  S )
 )
 
Theoremc0snmhm 38977* The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   &    |-  Z  =  ( 0g `  T )   =>    |-  ( ( S  e.  Mnd  /\  T  e.  Mnd  /\  B  =  { Z } )  ->  H  e.  ( T MndHom  S ) )
 
Theoremc0snghm 38978* The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   &    |-  Z  =  ( 0g `  T )   =>    |-  ( ( S  e.  Grp  /\  T  e.  Grp  /\  B  =  { Z } )  ->  H  e.  ( T  GrpHom  S ) )
 
Theoremzrrnghm 38979* The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
 ) )  ->  H  e.  ( T RngHomo  S )
 )
 
21.33.14.4  Ring homomorphisms (extension)
 
Theoremrhmfn 38980 The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
 |- RingHom  Fn  ( Ring  X.  Ring )
 
Theoremrhmval 38981 The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
 |-  (
 ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
 
Theoremrhmisrnghm 38982 Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.)
 |-  ( F  e.  ( R RingHom  S )  ->  F  e.  ( R RngHomo  S ) )
 
21.33.14.5  Ideals as non-unital rings
 
Theoremlidldomn1 38983* If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  ( U  e.  L  /\  U  =/=  {  .0.  } )  /\  I  e.  U )  ->  ( A. x  e.  U  ( ( I  .x.  x )  =  x  /\  ( x  .x.  I
 )  =  x ) 
 ->  I  =  .1.  ) )
 
Theoremlidlssbas 38984 The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( U  e.  L  ->  ( Base `  I )  C_  ( Base `  R )
 )
 
Theoremlidlbas 38985 A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( U  e.  L  ->  ( Base `  I )  =  U )
 
Theoremlidlabl 38986 A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  I  e.  Abel )
 
Theoremlidlmmgm 38987 The multiplicative group of a (left) ideal of a ring is a magma. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  (mulGrp `  I )  e. Mgm )
 
Theoremlidlmsgrp 38988 The multiplicative group of a (left) ideal of a ring is a semigroup. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  (mulGrp `  I )  e. SGrp )
 
Theoremlidlrng 38989 A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  I  e. Rng )
 
Theoremzlidlring 38990 The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  I  e.  Ring
 )
 
Theoremuzlidlring 38991 Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  U  e.  L ) 
 ->  ( I  e.  Ring  <->  ( U  =  {  .0.  }  \/  U  =  B ) ) )
 
Theoremlidldomnnring 38992 A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  ( U  e.  L  /\  U  =/=  {  .0.  } 
 /\  U  =/=  B ) )  ->  I  e/  Ring
 )
 
21.33.14.6  The non-unital ring of even integers
 
Theorem0even 38993* 0 is an even integer. (Contributed by AV, 11-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   =>    |-  0  e.  E
 
Theorem1neven 38994* 1 is not an even integer. (Contributed by AV, 12-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   =>    |-  1  e/  E
 
Theorem2even 38995* 2 is an even integer. (Contributed by AV, 12-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   =>    |-  2  e.  E
 
Theorem2zlidl 38996* The even integers are a (left) ideal of the ring of integers. (Contributed by AV, 20-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  U  =  (LIdeal ` ring )   =>    |-  E  e.  U
 
Theorem2zrng 38997* The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm 38877. (Contributed by AV, 20-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  U  =  (LIdeal ` ring )   &    |-  R  =  (rings  E )   =>    |-  R  e. Rng
 
Theorem2zrngbas 38998* The base set of R is the set of all even integers. (Contributed by AV, 31-Jan-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |-  E  =  ( Base `  R )
 
Theorem2zrngadd 38999* The group addition operation of R is the addition of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |- 
 +  =  ( +g  `  R )
 
Theorem2zrng0 39000* The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |-  0  =  ( 0g
 `  R )
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