Irreducible ring

In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways.
"Meet-irreducible" rings are referred to as "irreducible rings" in commutative algebra. This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed.
Meet-irreducible rings play an important part in commutative algebra, and directly irreducible and subdirectly irreducible rings play a role in the general theory of structure for rings. 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.
Definitions.
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.
The following conditions are equivalent for a commutative ring "R":
The following conditions are equivalent for a commutative ring "R":
The following conditions are equivalent for a ring "R":
The following conditions are equivalent for a ring "R":
Examples and properties.
If "R" is subdirectly irreducible or meet-irreducible, then it is also directly irreducible, but the converses are not true.
Generalizations.
Commutative meet-irreducible rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of an irreducible scheme.