In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a semi-local ring is a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
for which ''R''/J(''R'') is a
semisimple ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
, where J(''R'') is the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yie ...
of ''R''.
The above definition is satisfied if ''R'' has a finite number of maximal right ideals (and finite number of maximal left ideals). When ''R'' is a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
s".
Some literature refers to a commutative semi-local ring in general as a
''quasi-semi-local ring'', using semi-local ring to refer to a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
with finitely many maximal ideals.
A semi-local ring is thus more general than a
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 num ...
, which has only one maximal (right/left/two-sided) ideal.
Examples
* Any right or left
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are na ...
, any
serial ring, and any
semiperfect ring
In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exi ...
is semi-local.
* The quotient
is a semi-local ring. In particular, if
is a prime power, then
is a local ring.
* A finite direct sum of fields
is a semi-local ring.
* In the case of commutative rings with unity, this example is prototypical in the following sense: the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
shows that for a semi-local commutative ring ''R'' with unit and maximal ideals ''m
1, ..., m
n''
:
.
:(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩
i m
i=J(''R''), and we see that ''R''/J(''R'') is indeed a semisimple ring.
* The
classical ring of quotients for any commutative Noetherian ring is a semilocal ring.
* The
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
of an
Artinian module
In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if ...
is a semilocal ring.
* Semi-local rings occur for example in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
when a (commutative) ring ''R'' is
localized with respect to the multiplicatively closed subset ''S = ∩ (R \ p
i)'', where the ''p
i'' are finitely many
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s.
Textbooks
*
*
Ring theory
Localization (mathematics)
{{Abstract-algebra-stub