Perfect ring

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'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. 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 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 × , 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 is J. Furthermore R/J is a field, so that R is local, and R is right but not left perfect. 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.