Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-III
65
Theorem 1
Let M, N be R-modules and let f : M → N be a non-zero R-morphism. Then
1. If M is simple, f is a monomorphism.
2. If N is simple, f is an epimorphism.
Proof
1. Ker f is a submodule of M, since f is not the zero morphism, we must have ker f = (0), because M is
simple, so only submodules of M are (0) and M itself (if Ker f = M, then f(M) = (0) ⇒ f = 0, but
).
Hence Ker
is a monomorphism.
2. Imf is a submodule of N, But N is simple, so Imf
but
. Therefore,
Corollary : (Shur's Lemma) If M is a simple R-module, then the ring End of R-morphisms.
f : M –––→ M is a division ring.
Proof
From (1) and (2) above, every non-zero f ∈ End is an isomorphism and so is an invertible element in
the ring. Hence End R ( M ) is a division ring.
Fundamental structure theorem for finitely generated modules over a principal ideal domain :
Before proving this we have to build some tools needed to prove above theorem :
= f = ≠f
( 0
M
) =
)
N( = 0.
(),
f0
)
R( n M
)
Suppose we have a sequence of modules with a homomorphism from each module to the next :
f o f1
f 2 f 3
......... – → M – → M – → M – → ...........
This sequence is said to exact at M 1
if
Im f 1
= Ker f 2
o
The sequence is exact if it is exact at every module
An exact sequence of the form
α α β
( 0) – → M – → M – → M – → ( 0)
1 2
is called a short exact sequence.
1
2
or
Here f is an epimorphism.
Recall that every module over a general ring R is a homonorphic image of a free module. Every R-module M
forms part of a short exact sequence.
( 0) – → G – → F – → M – → ( 0)
where F is free, this is called a presentation of M; If M is finitely generated, F can be taken to be of finite
rank.
We shall use the result (without proving it)
"If R is a principal ideal domain, then for any integer n, any submodule of
. If
is free of rank at most n."
Using this, we assume that above G is free, at least when M is finitely generated. More precisely, when M is
generated by n elements, then it has a presentation
then