In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the field of commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, an irreducible ring is a commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

 A that satisfies one of the following equivalent conditions:

• A possesses exactly one minimal prime ideal;
• the zero ideal in A is irreducible
  Irreducible ideal
  In mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as a finite intersection of ideals properly containing it....
  
  , i.e. the intersection of two non-zero ideals of A always is non-zero;
• the spectrum
  Spectrum of a ring
  In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
  
   of A is irreducible.

Examples and non-examples

All integral domains are irreducible. More exactly, a commutative ring is integral if and only if it is both irreducible and reduced

Reduced ring

In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...

.

Non-examples are given by product rings such as Z × Z: here the intersection of the non-zero ideals {0} × Z and Z × {0} is equal to the zero ideal {0} × {0}.

Properties

Irreducible rings are connected

Connected ring

In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions:* A possesses no non-trivial idempotent elements;...

; in particular, their only idempotent elements are 0 and 1.

Generalizations

Irreducible rings play an elementary role in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, where this concept is generalized to the concept of an irreducible scheme.