Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Perfect ring
Summary
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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.