Generalizations of Perfect, Semiperfect, and Semiregular Rings (original) (raw)

On semi-projective modules and their endomorphism rings

Asian-European Journal of Mathematics, 2017

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

A decomposition theorem for -semisimple rings

Journal of Pure and Applied Algebra, 2004

A module M is said to satisfy the condition (˝ *) if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisÿes (˝ *) are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right˝ *-semisimple rings. Right˝ *-semisimple rings are right artinian. However, in general, a right˝ *-semisimple rings need not be left˝ *-semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left˝ *-semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.

Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Communications in Algebra, 2003

For a ring S, let K 0 (FGFl(S)) and K 0 (FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring R is semiperfect if and only if the group homomorphism K 0 (FGFl(R)) → K 0 (FGFl(R/J(R))) is an epimorphism and K 0 (FGFl(R)) = K 0 (FGPr(R)).

ON A RECENT GENERALIZATION OF SEMIPERFECT RINGS

Bulletin of the Australian Mathematical Society, 2008

It follows from a recent paper by Ding and Wang that any ring which is generalized supplemented as left module over itself is semiperfect. The purpose of this note is to show that Ding and Wang's claim is not true and that the class of generalized supplemented rings lies properly between the class of semilocal and semiperfect rings. Moreover we rectify their "theorem" by introducing a wider notion of local submodules.

Rings for which every cosingular module is projective abstract

Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica

Let R be a ring and M be an R-module. We say that M is (SCP) CP provided every (simple) cosingular R-module is M-projective. We give some equivalent conditions for a (SCP) CP-module. We show that every simple cosingular R-module is projective if and only if R is a GV -ring (GCO-ring). Let R be a right perfect ring. We prove that, R is a CP-ring if and only if every R-module is a direct sum of a non-cosingular module and a semisimple module. We also give a condition for a Harada ring to be a CP-ring.

A characterization of perfect rings

Pacific Journal of Mathematics, 1970

has shown that if a ring R is right perfect, then a certain torsion in the category Mod R of left ϋί-modules is closed under taking direct products. Extending his method, J. S. Alin and E. P. Armendariz showed later that this is true for every (hereditary) torsion in Mod R. Here, we offer a very simple proof of this result. However, the main purpose of this paper is to present a characterization of perfect rings along these lines: A ring R is right perfect if and only if every (hereditary) torsion in Mod R is fundamental (i.e., derived from "prime" torsions) and closed under taking direct products; in fact, then there is a finite number of torsions, namely 2 n for a natural number n. Finally, examples of rings illustrating that the above characterization cannot be strengthened are provided. Thus, an example of a ring R± is given which is not perfect, although there are only fundamental torsions in Mod Ri, and only 4 = 2 2 of these. Furthermore, an example of a ring R 2 * is given which is not perfect and which, at the same time, has the property that there is only a finite number (namely, 3) of (hereditary) torsions in Mod i? 2 * all of which are closed under taking direct products. Moreover, the ideals of R 2 * form a chain (under inclusion) and Rad R 2 * is a nil idempotent ideal; all the other proper ideals are nilpotent and R 2 * can be chosen to have a (unique) minimal ideal.

On a class of perfect rings

Canadian Journal of Mathematics, 1970

In [3], the perfect rings of Bass [1] were characterized in terms of torsions in the following way: A ring R is right perfect if and only if every (hereditary) torsion in the category Mod R of all left R-modules is fundamental (i.e. generated by some minimal torsions) and closed under taking direct products; as a consequence, the number of all torsions in Mod R is finite and equal to 2 n for a natural n. Here, we present a simple description of those rings R which allow only two (trivial) torsions, viz. 0 and Mod R (and thus, are right perfect by [3]). Finite direct sums of these rings represent a natural generalization of completely reducible (i.e. artinian semisimple) rings (cf. Theorem 2) and we shall call them for that matter π-reducible rings. They can also be characterized in terms of their idempotent two-sided ideals.