In the area of abstract algebra known as ring theory, 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 1960).
A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.
The following equivalent definitions of a left perfect ring R are found in (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 ℕ× ℕ, and which only have finitely many nonzero entries above the diagonal, and denote this set by
J. Also take the matrix
I with all 1's on the diagonal, and form the set
R = { f ⋅ I + j ∣ f ∈ F , j ∈ J } 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. (Lam 2001, p.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.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, ..., en of idempotents with each ei R ei 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.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.
