Irreducible ring

• A irreducible ring is one in which the intersection of two nonzero ideals is always nonzero.
• A directly irreducible ring is ring which cannot be written as the direct sum of two nonzero rings.
• A subdirectly irreducible ring is a ring with a unique, nonzero minimum two-sided ideal.

Definitions

• 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.

• R possesses exactly one minimal prime ideal ;
• the spectrum of R is irreducible.

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

• 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

• All integral domains are meet-irreducible and subdirectly irreducible. In fact, a commutative ring is a domain if and only if it is both meet-irreducible and reduced.
• The quotient ring Z/ is a ring which has all three senses of irreducibility, but it is not a domain. Its only proper ideal is /, which is maximal, hence prime. The ideal is also minimal.
• The direct product of two nonzero rings is never directly irreducible, and hence is never meet-irreducible or subdirectly irreducible. For example, in Z × Z the intersection of the non-zero ideals × Z and Z × is equal to the zero ideal × .
• Commutative directly irreducible rings are connected rings; that is, their only idempotent elements are 0 and 1.

  Generalizations