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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 irreducibe 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.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:

R is meet-irreducible;
the zero ideal in R is irreducible, i.e. the intersection of two non-zero ideals of A always is non-zero.

The following conditions are equivalent for a commutative ring R:

R possesses exactly one minimal prime ideal (this prime ideal may be the zero ideal);
the spectrum of R is irreducible.

The following conditions are equivalent for a ring R:

R is directly irreducible;
R has no central idempotents except for 0 and 1.

The following conditions are equivalent for a ring R:

R is subdirectly irreducible;
when R is written as a subdirect product of rings, then one of the projections of R onto a ring in the subdirect product is an isomorphism;
The intersection of all nonzero ideals of R is nonzero.

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.

Mathematics Encyclopedia

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