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 (that is, not equal to 1 or 0) idempotent elementss;

• the spectrum of A with the Zariski topology is a connected space.

Examples and non-examples

Connectedness defines a fairly general class of commutative rings. For example, all local rings and all (meet-)irreducible rings are connected. In particular, all integral domains are connected. Non-examples are given by product rings such as Z×Z; here the element (1,0) is a non-trivial idempotent.

Generalizations

In algebraic geometry, connectedness is generalized to the concept of a connected scheme.

References