Perfect ring

{{no footnotes|date=March 2016}} {{about|perfect rings as introduced by Hyman Bass|perfect rings of characteristic p generalizing perfect fields|perfect field}}

In the area of abstract algebra known as ring theory, a '''left perfect ring''' is a type of ring over 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.{{sfn|Bass|1960}}

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 Anderson and Fuller:{{sfn|Anderson|Fuller|1992|p=315}} * Every left ''R''-module has a projective cover. * ''R''/J(''R'') is semisimple 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 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. * ''R'' contains no infinite orthogonal set of idempotents, and every non-zero right ''R''-module contains a minimal submodule.

===Examples=== * Right or left Artinian rings, and semiprimary rings are known to be right-and-left perfect. * The following is an example (due to Bass) of a local ring 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 <math>\mathbb{N} \times \mathbb{N}</math>, and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by <math>J</math>. Also take the matrix <math>I\,</math> with all 1's on the diagonal, and form the set ::<math>R = \{f\cdot I+j\mid f\in F, j\in J \}\,</math> :It can be shown that ''R'' is a ring with identity, whose Jacobson radical is ''J''. Furthermore ''R''/''J'' is a field, so that ''R'' is local, and ''R'' is right but not left perfect.{{sfn|Lam|2001|pp=345-346}}

===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 holds for projective modules. {{Citation needed|date=July 2011}}

==Semiperfect ring== ===Definition=== Let ''R'' be ring. Then ''R'' is semiperfect if any of the following equivalent conditions hold:

* ''R''/J(''R'') is semisimple and idempotents lift modulo J(''R''), where J(''R'') is the Jacobson radical of ''R''. * ''R'' has a complete orthogonal set ''e''1, ..., ''e''''n'' of idempotents with each ''eiRei'' a local ring. * 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 <math>R</math>-modules is Krull-Schmidt.

===Examples===

Examples of semiperfect rings include: * Left (right) perfect rings. * Local rings. * Left (right) Artinian rings. * 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 to a semiperfect ring is also semiperfect.

===Basic ring===

For a fixed semiperfect ring ''R'', there is (up to a ring isomorphism) a canonical representative of the class of rings Morita equivalent to ''R'' called a '''basic ring for ''R''.''' A semiperfect ring is basic if and only if ''R''/J(''R'') is a direct product of division rings.

Given a complete orthogonal set ''e''1, ..., ''e''''n'' of local idempotents, we can assume ordering of this set in such a way that the for some ''i≤n'' the right modules ''e''1R, ..., ''e''''i''R form a complete set of finitely generated projective right modules. The idempotent ''e''=''e''1+ ... + ''e''''i'' is then called a basic idempotent and the corner ring ''B:=eRe'' isomorphic to ''End''R(''e''1R + ...+ ''e''''i'' ''R'') is the basic ring for ''R''.

The map ''I''→''IR'' defines embedding of the lattice of right ideals of ''B'' into that of ''R''. The map ''I''→''RIR'' defines an isomorphism of the lattice of ideals of ''B'' and the lattice of ideals of ''R''. This isomorphism respects multiplication of ideals. {{sfn|Lam|2001|pp=371-375}}

==References== {{reflist}}

==Further reading== {{refbegin}} * {{Citation|last1=Anderson|first1=Frank W|last2=Fuller|first2=Kent R|title=Rings and Categories of Modules|publisher=Springer-Verlag|year=1992|edition=2nd|isbn=978-0-387-97845-1|url=https://www.springer.com/gp/book/9780387978451}} * {{Citation|last1=Bass|first1=Hyman|title=Finitistic dimension and a homological generalization of semi-primary rings|doi=10.2307/1993568|jstor=1993568|mr=0157984|year=1960|journal=Transactions of the American Mathematical Society|issn=0002-9947|volume=95|issue=3|pages=466–488|doi-access=free}} * {{Citation|last=Lam|first=T. Y.|author-link=Tsit Yuen Lam|title=A first course in noncommutative rings|series=Graduate Texts in Mathematics|volume=131|edition=2|publisher=Springer-Verlag|place=New York|year=2001|isbn=0-387-95183-0|mr=1838439|doi=10.1007/978-1-4419-8616-0}} {{refend}}

Category:Ring theory