semiprimary ring
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Author(s):  
Truong Cong Quynh ◽  
Adel Abyzov ◽  
Dinh Duc Tai

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module [Formula: see text] over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of [Formula: see text] extend to nilpotent endomorphisms of [Formula: see text]. It is also shown that a right module [Formula: see text] over a prime right Goldie ring with [Formula: see text] is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if [Formula: see text] is a finitely generated nilpotent-coinvariant right module with [Formula: see text] square-full, then [Formula: see text] is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.


1989 ◽  
Vol 40 (2) ◽  
pp. 189-197 ◽  
Author(s):  
Vlastimil Dlab

The concept of the heredity measure of a semiprimary ring (or finite-dimensional algebra) is introduced and some of its elementary properties are studied.


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