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 elements; 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.
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