In the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
known as
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, 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 exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book.
A semiperfect ring is a ring over which every
finitely generated left module has a projective cover. This property is left-right symmetric.
Perfect ring
Definitions
The following equivalent definitions of a left perfect ring ''R'' are found in Aderson and Fuller:
* Every left ''R'' module has a projective cover.
* ''R''/J(''R'') is
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 ...
and J(''R'') is left T-nilpotent (that is, for every infinite sequence of elements of J(''R'') there is an ''n'' such that the product of first ''n'' terms are zero), 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 definitio ...
of ''R''.
* (Bass' Theorem P) ''R'' satisfies the
descending chain condition on principal right ideals. (There is no mistake; this condition on ''right'' principal ideals is equivalent to the ring being ''left'' perfect.)
* Every
flat left ''R''-module is
projective.
* ''R''/J(''R'') is semisimple and every non-zero left ''R'' module contains a
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 ...
.
* ''R'' contains no infinite orthogonal set of
idempotent
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, and every non-zero right ''R'' module contains a minimal submodule.
Examples
* 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 n ...
s, and
semiprimary rings are known to be right-and-left perfect.
* The following is an example (due to Bass) of 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 ...
which is right but not left perfect. Let ''F'' be a field, and consider a certain ring of
infinite matrices over ''F''.
:Take the set of infinite matrices with entries indexed by
×
, and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by
. Also take the matrix
with all 1's on the diagonal, and form the set
:
:It can be shown that ''R'' is a ring with identity, whose
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 ...
is ''J''. Furthermore ''R''/''J'' is a field, so that ''R'' is local, and ''R'' is right but not left perfect.
Properties
For a left perfect ring ''R'':
* From the equivalences above, every left ''R'' module has a maximal submodule and a projective cover, and the flat left ''R'' modules coincide with the projective left modules.
* An analogue of the
Baer's criterion
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule of ...
holds for projective modules.
Semiperfect ring
Definition
Let ''R'' be ring. Then ''R'' is semiperfect if any of the following equivalent conditions hold:
* ''R''/J(''R'') is
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 ...
and
idempotent
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 lift modulo J(''R''), 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 definitio ...
of ''R''.
* ''R'' has a complete orthogonal set ''e''
1, ..., ''e''
''n'' of idempotents with each ''e''
''i'' ''R e''
''i'' 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 ...
.
* Every
simple left (right)
''R''-module has a
projective cover.
* Every
finitely generated left (right) ''R''-module has a projective cover.
* The category of finitely generated projective
-modules is
Krull-Schmidt.
Examples
Examples of semiperfect rings include:
* Left (right) perfect rings.
*
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 ...
s.
*
Kaplansky's theorem on projective modules In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called ''local'' if for each element ''x'', eit ...
* Left (right)
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 n ...
s.
*
Finite dimensional ''k''-algebras.
Properties
Since a ring ''R'' is semiperfect iff every
simple left
''R''-module has a projective cover, every ring
Morita equivalent
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of module ...
to a semiperfect ring is also semiperfect.
Citations
References
*
*
*
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Ring theory