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...

, a connected 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 no non-trivial (that is, not equal to 1 or 0) idempotent elements;
• 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 a connected space
  Connected space
  In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
  
  .

Examples and non-examples

Connectedness defines a fairly general class of commutative rings. For example, all local ring

Local ring

In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...

s and all irreducible ring

Irreducible ring

In mathematics, especially in the field of commutative algebra, an irreducible ring is a commutative ring A that satisfies one of the following equivalent conditions:* A possesses exactly one minimal prime ideal;...

s 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

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...

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