Irreducible module

simple module

A non-zero unital module $M$ over a unital ring $R$ that contains only two submodules: the null module and $M$ itself.

Examples. 1) If $R = \mathbf{Z}$ is the ring of integers, then the irreducible $R$-modules are the Abelian groups of prime order. 2) If $R$ is a skew-field, then the irreducible $R$-modules are the one-dimensional vector spaces over $R$. 3) If $D$ is a skew-field, $V$ is a left vector space over $D$ and $R = \End_D(V)$ is the ring of linear transformations of $V$ (or a dense subring of it), then the right $R$-module $V$ is irreducible. 4) If $G$ is a group and $k$ is a field, then the irreducible representations of $G$ over $K$ are precisely the irreducible modules over the group algebra $R = k[G]$.

A right $R$-module $M$ is irreducible if and only if it is isomorphic to $R/I$, where $I$ is a maximal right ideal in $R$. If $A$ and $B$ are irreducible $R$-modules and $f \in \Hom_R(A,B)$, then either $f=0$ or $f$ is an isomorphism (which implies that the endomorphism ring of an irreducible module is a skew-field). If $R$ is an algebra over an algebraically closed field $K$ and if $A$ and $B$ are irreducible modules over $R$, then (Schur's lemma) $$ \Hom_R(A,B) = \begin{cases} K & \ \text{if}\ A \cong B\ ; \\ 0 & \ \text{otherwise} \ .\end{cases} $$

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.

References

[1]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[3]	J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)
[4]	C. Faith, "Algebra: rings, modules, and categories" , 1–2 , Springer (1973–1976)