Irreducible ring

This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed.

Subdirectly irreducible algebras have also found use in number theory.

This article follows the convention that rings have multiplicative identity, but are not necessarily commutative.

The terms "meet-reducible", "directly reducible" and "subdirectly reducible" are used when a ring is not meet-irreducible, or not directly irreducible, or not subdirectly irreducible, respectively.

Commutative meet-irreducible rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of an irreducible scheme.