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:
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.