In
mathematics, especially in the field of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, 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 element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s;
* the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of ''A'' with the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
is a
connected space.
Examples and non-examples
Connectedness defines a fairly general class of commutative rings. For example, all
local ring In abstract algebra, more specifically 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 n ...
s and all (meet-)
irreducible ring
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways.
* A (meet-)irreducible ring is a ring in which the intersection of two non-zero ideals is always non-zero.
* A directly irreducib ...
s are connected. In particular, all
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
s 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
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.
...
.
References
*
Commutative algebra
Ring theory
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