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.

Perfect ring

Definitions

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.

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 [math]\displaystyle{ \mathbb{N} \times \mathbb{N} }[/math], and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by [math]\displaystyle{ J }[/math]. Also take the matrix [math]\displaystyle{ I\, }[/math] with all 1's on the diagonal, and form the set

[math]\displaystyle{ R = \{f\cdot I+j\mid f\in F, j\in J \}\, }[/math]

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 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, ..., e_n of idempotents with each e_iRe_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 [math]\displaystyle{ R }[/math]-modules is Krull-Schmidt.

Examples

Examples of semiperfect rings include:

• Left (right) perfect rings.
• Local rings.
• Kaplansky's theorem on projective modules
• 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.

References

• Anderson, Frank W; Fuller, Kent R (1992), Rings and Categories of Modules (2nd ed.), Springer-Verlag, ISBN 978-0-387-97845-1, https://www.springer.com/gp/book/9780387978451 
• Bass, Hyman (1960), "Finitistic dimension and a homological generalization of semi-primary rings", Transactions of the American Mathematical Society 95 (3): 466–488, doi:10.2307/1993568, ISSN 0002-9947 
• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, 131 (2 ed.), New York: Springer-Verlag, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0