A nonzero module 
over a ring 
whose only submodules are the
module itself and the zero module. It is also called
a simple module, and in fact this is the name more frequently used nowadays
(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), which means that 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 
are 
, 
and 
, so 
is not irreducible, whereas it is certainly meet-irreducible.
This ambiguity in terminology is solved in the context of rings, since a simple ring is a ring that is irreducible as a module over itself, whereas an irreducible ring is a ring which is meet-irreducible as a module over itself.
Irreducible modules play a crucial role in representation theory.
