A non-zero
unitary module
over a ring
with a unit element that contains only two submodules: the null module and
itself.
Examples.
1) If
is the ring of integers, then the irreducible
-modules
are the Abelian groups of prime order. 2) If
is a skew-field, then the irreducible
-modules
are the one-dimensional vector spaces over
.
3) If
is a skew-field,
is a left vector space over
and
is the ring of linear transformations of
(or a dense subring of it), then the right
-module
is irreducible. 4) If
is a group and
is a field, then the irreducible representations of
over
are precisely the irreducible modules over the
group algebra
.
A right
-module
is irreducible if and only if it is isomorphic to
,
where
is a maximal right ideal in
.
If
and
are irreducible
-modules
and
,
then either
or
is an isomorphism (which implies that the endomorphism ring
of an irreducible module is a skew-field). If
is an algebra over an algebraically closed field and if
and
are irreducible modules over
,
then
(Schur's lemma)
The concept of an irreducible module is fundamental in the theories
of rings and group representations. By means of it one defines the
composition sequence
and the
socle
of a module, the
Jacobson radical
of a module and of a ring, and a
completely-reducible module.
Irreducible modules are involved in the definition of a
number of important classes of rings: classical
semi-simple rings, primitive rings, and others (cf.
Classical semi-simple ring;
Primitive ring).