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In the area ofabstract algebra known asring theory, aleft perfect ring is a type ofring over which all leftmodules haveprojective 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 inBass's book.^[1]

Asemiperfect ring is a ring over which everyfinitely generated left module has a projective cover. This property is left-right symmetric.

The following equivalent definitions of a left perfect ringR are found in Anderson and Fuller:^[2]

• Every leftR-module has a projective cover.
• R/J(R) issemisimple and J(R) isleft T-nilpotent (that is, for every infinite sequence of elements of J(R) there is ann such that the product of firstn terms are zero), where J(R) is theJacobson radical ofR.
• (Bass' Theorem P)R satisfies thedescending chain condition onprincipal right ideals. (There is no mistake; this condition onright principal ideals is equivalent to the ring beingleft perfect.)
• Everyflat leftR-module isprojective.
• R/J(R) is semisimple and everynon-zero leftR-module contains amaximal submodule.
• R contains no infiniteorthogonal set ofidempotents, and every non-zero rightR-module contains aminimal submodule.

• Right or leftArtinian rings, andsemiprimary rings are known to be right-and-left perfect.
• The following is an example (due to Bass) of alocal ring which is right but not left perfect. LetF be afield, and consider a certain ring ofinfinite matrices overF.

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

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

It can be shown thatR is a ring with identity, whose Jacobson radical isJ. FurthermoreR/J is a field, so thatR is local, andR is right but not left perfect.^[3]

For a left perfect ringR:

• From the equivalences above, every leftR-module has a maximal submodule and a projective cover, and the flat leftR-modules coincide with the projective left modules.
• An analogue of theBaer's criterion holds for projective modules.^{[citation needed]}

LetR be ring. ThenR is semiperfect if any of the following equivalent conditions hold:

• R/J(R) issemisimple andidempotentslift modulo J(R), where J(R) is theJacobson radical ofR.
• R has a complete orthogonal sete₁, ...,e_n of idempotents with eache_iRe_i alocal ring.
• Everysimple left (right)R-module has aprojective cover.
• Everyfinitely generated left (right)R-module has a projective cover.
• Thecategory of finitely generatedprojective${\displaystyle R}$-modules isKrull-Schmidt.

Examples of semiperfect rings include:

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

Since a ringR is semiperfect iff every simple leftR-module has a projective cover, every ringMorita equivalent to a semiperfect ring is also semiperfect.

• Ring theory

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