An irreducible module is a nonzero module 
over a ring 
whose only submodules
are the module itself and the zero
module. It is also called a simple module (Rowen 1988). Behrens' (1972,
p. 23) definition includes the additional condition that 
be not the zero module.
Sometimes, the term irreducible is used as an abbreviation for meet-irreducible (Kasch 1982, p. 161), which means a nonzero module in which the intersection of two nonzero submodules is always nonzero.
These two irreducibility notions are different: every irreducible module is meet-irreducible, but the converse does not hold. For example, the submodules
of the 
-module 
are 
, 
, and 
, so 
is not irreducible, whereas it is meet-irreducible.
For a ring, the corresponding terminology is expressed using two-sided ideals. A simple ring has no proper nonzero two-sided ideals, whereas in an irreducible ring the intersection of two nonzero two-sided ideals is always nonzero.
Irreducible modules play a crucial role in representation theory.
