In the context of a
module ''M'' over a
ring
(The) Ring(s) 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
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
''R'', the top of ''M'' is the largest
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
quotient module
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups ...
of ''M'' if it exists.
For finite-dimensional ''k''-algebras (''k'' a field) R, if rad(''M'') denotes the intersection of all proper
maximal submodule
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 ...
s of ''M'' (the
radical of the module), then the top of ''M'' is ''M''/rad(''M''). In the case of local rings with maximal ideal ''P'', the top of ''M'' is ''M''/''PM''. In general if ''R'' is a
semilocal ring (=semi-artinian ring), that is, if ''R''/Rad(''R'') is an
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 ...
, where Rad(''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 definitio ...
of ''R'', then ''M''/rad(''M'') is a semisimple module and is the top of ''M''. This includes the cases of local rings and finite dimensional algebras over fields.
See also
*
Projective cover In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes.
Definition
...
*
Radical of a module In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle ...
*
Socle (mathematics)
In mathematics, the term socle has several related meanings.
Socle of a group
In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgroup generated by the minimal normal subgroups of ''G''. It can happen that a ...
References
*
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now k ...
, ''Commutative algebra with a view toward Algebraic Geometry''
Commutative algebra
Module theory
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