Perfect ring

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. A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.

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. 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: 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 \mathbb{N} \times \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\, with all 1's on the diagonal, and form the set R = \{f\cdot I+j\mid f\in F, j\in 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. ===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. ==Semiperfect ring== ===Definition=== Let R be a 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 e i Re 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=== 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 1 R, ..., 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 1 R + ...+ 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. ==References== ==Further reading==

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