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.[1]

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

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Definitions

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The following equivalent definitions of a left perfect ring R are found in Aderson and Fuller:[2]

Examples

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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.[3]

Properties

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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. [citation needed]

Semiperfect ring

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Definition

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Let R be ring. Then R is semiperfect if any of the following equivalent conditions hold:

Examples

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Examples of semiperfect rings include:

Properties

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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.

Citations

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  1. ^ Bass 1960.
  2. ^ Anderson & Fuller 1992, p. 315.
  3. ^ Lam 2001, pp. 345–346.

References

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  • Anderson, Frank W; Fuller, Kent R (1992), Rings and Categories of Modules (2nd ed.), Springer-Verlag, ISBN 978-0-387-97845-1
  • 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, JSTOR 1993568, MR 0157984
  • Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439