In the area of abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

 known as ring theory

Ring theory

In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

, a left perfect ring is a type of ring in which all left modules have projective cover

Projective cover

In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.- Definition :...

s. 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 .

Definitions

The following equivalent definitions of a left perfect ring R are found in :

• Every left R module has a projective cover.
• R/J(R) is semisimple
  Semisimple module
  In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring which is a semisimple module over itself is known as an artinian semisimple ring...
  
   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
  Jacobson radical
  In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
  
   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
  Flat module
  In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
  
   left R-module is projective
  Projective module
  In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
  
  .
• 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 ring
  Artinian ring
  In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
  
  s, and semiprimary ring
  Hopkins–Levitzki theorem
  In the branch of abstract algebra called ring theory, the Akizuki-Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R is called semiprimary if R/J is semisimple and J is a nilpotent ideal, where J denotes the...
  
  s are known to be right-and-left perfect.
• The following is an example (due to Bass) of a local ring
  Local ring
  In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
  
   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 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

Jacobson radical

In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...

 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.
• R is a semiperfect ring
  Semiperfect ring
  In abstract algebra, a semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left right symmetric.- Definition :Let R be ring...
  
  , since one of the characterizations of semiperfect rings is: "All finitely generated left R modules have projective covers."
• An analogue of the Baer's criterion holds for projective modules.