Perfect ring#Semiperfect ring

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.

• 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 $\backslash mathbb\{N\}\; \backslash times\; \backslash mathbb\{N\}$, and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by $J$. Also take the matrix $I\backslash ,$ with all 1's on the diagonal, and form the set

::$R\; =\; \backslash \{f\backslash cdot\; I+j\backslash mid\; f\backslash in\; F,\; j\backslash in\; J\; \backslash \}\backslash ,$

: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}}

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}}

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 e₁, ..., e_n of idempotents with each e_iRe_i 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 $R$-modules is Krull-Schmidt.

Examples of semiperfect rings include:

• Left (right) perfect rings.
• Local rings.
• Kaplansky's theorem on projective modules
• Left (right) Artinian rings.
• Finite dimensional k-algebras.

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.

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• {{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.|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}}

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Category:Ring theory