When is a semiperfect ring right PF? (English) Zbl 1183.16003
A ring \(R\) is called right PF if every faithful right \(R\)-module is a generator for the category of right \(R\)-modules. It is well-known that a ring \(R\) is right PF if and only if \(R\) is semiperfect and right self-injective with essential right socle. A module \(V\) is called local if it has a largest proper submodule, while a module \(W\) is called colocal if \(\text{Soc}(W)\) is simple and essential in \(W\).
The authors show that a ring \(R\) is right PF if and only if \(R_R\) has finite uniform dimension and every colocal, injective right \(R\)-module is projective. As a consequence a semiperfect ring \(R\) is right PF if and only if the injective hull of every simple right \(R\)-module is projective.
The authors show that a ring \(R\) is right PF if and only if \(R_R\) has finite uniform dimension and every colocal, injective right \(R\)-module is projective. As a consequence a semiperfect ring \(R\) is right PF if and only if the injective hull of every simple right \(R\)-module is projective.
Reviewer: J. K. Park (Pusan)
MSC:
16D50 | Injective modules, self-injective associative rings |
16L30 | Noncommutative local and semilocal rings, perfect rings |
16D80 | Other classes of modules and ideals in associative algebras |
16L60 | Quasi-Frobenius rings |
16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
Keywords:
semiperfect rings; right PF-rings; colocal modules; self-injective rings; right pseudo-Frobenius ringsReferences:
[1] | DOI: 10.1007/978-1-4612-4418-9 · doi:10.1007/978-1-4612-4418-9 |
[2] | DOI: 10.1090/S0002-9939-97-03747-7 · Zbl 0871.16012 · doi:10.1090/S0002-9939-97-03747-7 |
[3] | Faith C., Grundl. Math. Wiss. 191 |
[4] | DOI: 10.1142/S0219498802000070 · Zbl 1034.16005 · doi:10.1142/S0219498802000070 |
[5] | Osofsky B. L., J. Algebra 14 pp 373– · Zbl 1115.15006 |
[6] | Thoang L. D., Acta Math. Univ. Comenianae pp 199– |
[7] | DOI: 10.4153/CMB-2005-029-5 · Zbl 1089.16005 · doi:10.4153/CMB-2005-029-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.