文档介绍：page 1 of Chapter 9
CHAPTER 9 INTRODUCING MUTATIVE ALGEBRA
We will discuss mutative rings and their modules, concentrating on two funda-
mental results, the Wedderburn structure theorem and Maschke’s theorem. Further insight
into the structure of rings will be provided by the Jacobson radical.
Semisimple Modules
Avector space is the direct sum of one-dimensional subspaces (each subspace consists
of scalar multiples of a basis vector). Aone-dimensional space is simple in the sense that it
does not have a nontrivial proper subspace. Thus any vector space is a direct sum of simple
subspaces. We examine those modules which behave in a similar manner.
Definition An R-module M is simple if M = 0 and the only submodules of M are
0 and M.
Theorem Let M be a nonzero R-module. The following conditions are equivalent,
and a module satisfying them is said to be semisimple pletely reducible.
(a) M is a sum of simple modules;
(b) M is a direct sum of simple modules;
(c) If N is a submodule of M, then N is a direct summand of M, that is, there is a submodule
N of M such that M = N ⊕ N .
Proof.
∈⊆
(a) implies (b). Let M be the sum of simple modules Mi,i I, denote j∈J Mj
by M(J). By Zorn’s lemma, there is a maximal subset J of I such that the sum defining
N = M(J) is direct. We will show that M = N. First assume that i/∈ J. Then N ∩ Mi is
a submodule of the simple module Mi, so it must be either 0 or ∩ Mi = 0, then
M(J ∪{i}) is direct, contradicting maximality of ∩ Mi = Mi,soMi ⊆ N. But
if i ∈ J, then Mi ⊆ N by definition of N. Therefore Mi ⊆ N for all i, and since M is the
sum of all the Mi,wehaveM = N.
(b) implies (c). This is essentially the same as (a) implies (b). Let N be a submodule of M,
where M is the direct sum of simple modules Mi,i∈ I. Let J be a maximal subset of I such
that the sum N + M(J) is direct. If i/∈ J then exactly as before, Mi ∩(N ⊕ M(J)) = Mi,
so Mi ⊆ N ⊕ M(J). This holds for i ∈ J as