³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980714889³2ip³2129.97.13.28³2data³2newauthor³31³3minor³30³3text³3A left R-module consists of some commutative group (M,+) together with a RinG of scalars (R,+,*), together with an operation RxM->M (scalar multiplication, usually just denoted *) such that

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry

A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an EndoMorphism of M.  The set of all EndoMorphism''s'' of M is denoted End(M) and forms a RinG under addition and composition, so the above actually defines a HomoMorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module under the IntegerNumbers, and is either faithful under them or some ModularArithmetic.

A module under a field is called a VectorSpace.
³3summary³3*³2tscreate³2980714889³2keepts³2980715049³2id³21076³2host³2cnts6p28.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980715049³2ip³2129.97.13.28³2data³2text³3A left R-module consists of some commutative group (M,+) together with a RinG of scalars (R,+,*), together with an operation RxM->M (scalar multiplication, usually just denoted *) such that

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry

A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an EndoMorphism of M.  The set of all EndoMorphism''s'' of M is denoted End(M) and forms a RinG under addition and composition, so the above actually defines a HomoMorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module under the IntegerNumbers, and is either faithful under them or some ModularArithmetic.

A module under a FielD is called a VectorSpace.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980714889³2keepts³2980715150³2id³21076³2host³2cnts6p28.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980715150³2ip³2129.97.13.28³2data³2newauthor³30³3minor³31³3text³3A left R-module consists of some commutative group (M,+) together with a RinG of scalars (R,+,*), together with an operation R x M->M (scalar multiplication, usually just denoted *) such that

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry

A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an EndoMorphism of M.  The set of all EndoMorphism''s'' of M is denoted End(M) and forms a RinG under addition and composition, so the above actually defines a HomoMorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module under the IntegerNumbers, and is either faithful under them or some ModularArithmetic.

A module under a FielD is called a VectorSpace.
³3summary³3*³2tscreate³2980714889³2keepts³2980715411³2id³21076³2host³2cnts6p28.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980715411³2ip³2129.97.13.28³2data³2text³3A left R-module consists of some commutative group (M,+) together with a RinG of scalars (R,+,*), together with an operation R x M->M (scalar multiplication, usually just denoted *) such that

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry

A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an EndoMorphism of M.  The set of all EndoMorphism''s'' of M is denoted End(M) and forms a RinG under addition and composition, so the above actually defines a HomoMorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module under the IntegerNumbers, and is either faithful under them or some ModularArithmetic.

Another thing to note is that End(M), treated as a group, is also a module under R in a natural way.  When R is a FielD, this constitutes an AssociativeAlgebra.  Module under FielD''s'' are called VectorSpace''s''.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980714889³2keepts³2980718029³2id³21076³2host³2cnts6p28.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980718029³2ip³2129.97.112.13³2data³2newauthor³30³3minor³31³3text³3A left R-module consists of some commutative GrouP (M,+) together with a RinG of scalars (R,+,*), together with an operation R x M->M (scalar multiplication, usually just denoted *) such that

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry

A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an EndoMorphism of M.  The set of all endomorphisms of M is denoted End(M) and forms a ring under addition and composition, so the above actually defines a HomoMorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module under the IntegerNumbers, and is either faithful under them or some ModularArithmetic.

Another thing to note is that End(M), treated as a group, is also a module under R in a natural way.  When R is a FielD, this constitutes an AssociativeAlgebra.  Module under FielD''s'' are called VectorSpace''s''.
³3summary³3*³2tscreate³2980714889³2keepts³2980718471³2id³21076³2host³2cnts1p13.uwaterloo.ca³2revision³25