Simple module
In
abstract algebra, a (left or right)
module S over a
ring R is called
simple or
irreducible if it is not the
zero module 0 and if its only
submodules are 0 and
S. Understanding the simple modules over a ring is usually helpful because these modules form the "building blocks" of all other modules in a certain sense.
Abelian groups are the same as
Z-modules. The simple
Z-modules are precisely the
cyclic groups of
prime order.
If
K is a
field and
G is a
group, then a
group representation of
G is a
left module over the
group ring KG. The simple
KG modules are also known as
irreducible representations. A major aim of
representation theory is to list those irreducible representations for a given group.
Given a ring
R and a
left ideal I in
R then
I is a simple
R-module if and only if
I is a minimal left ideal in
R (does not contain any other non trivial left ideals). The
factor module R /
I is a simple
R-module if and only if
I is a maximal left ideal in
R (is not contained in any other non-trivial left ideals).
The simple modules are precisely the modules of
length 1; this is a reformulation of the definition.
Every simple module is
indecomposable, but the converse is in general not true.
Every simple module is
cyclic, that is it is generated by one element
Not every module has a simple submodule; consider for instance the
Z-module
Z in light of the first example above.
If
S is a simple module and
f :
S →
T is a
module homomorphism, then
f is either the zero homomorphism or
injective. This is because the
kernel of
f is a submodule of
S and thus is, by the definition of a simple module, either 0 or
S. If
T is also a simple module, then
f is either zero or an
isomorphism. This is because the
image of
f is a submodule of
T and thus is either 0 or
T. Taken together, this implies that the
endomorphism ring of any simple module is a
division ring. This result is known as
Schur's lemma.
The converse of Schur's lemma is not true in general. For example, the
Z-module
Q is not simple, but its endomorphism ring is isomorphic to the field
Q.
*
semisimple modules are modules that can be written as a sum of simple submodules
*
simple groups are similarly defined to simple modules
