Modules which are coinvariant under automorphisms of their projective covers

Abstract

In this paper we study modules coinvariant under automorphisms of their projective covers. We first provide an alternative, and in fact, a more succinct and conceptual proof for the result that a module M is invariant under automorphisms of its injective envelope if and only if given any submodule N of M , any monomorphism f : N → M can be extended to an endomorphism of M and then, as a dual of it, we show that over a right perfect ring, a module M is coinvariant under automorphisms of its projective cover if and only if for every submodule N of M , any epimorphism ϕ : M → M/N can be lifted to an endomorphism of M .2010 Mathematics Subject Classification. 16D40, 16D80.

On image summand coinvariant modules and kernel summand invariant modules

“…Moreover, Singh and Srivastava [22] introduced a dual notion of an automorphism invariant module and proved that a lifting module over a right perfect ring is dual automorphism invariant if and only if it is quasi-projective. After that, Guil Asensio et al [9] showed that a module over a right perfect ring is dual automorphism invariant if and only if it is pseudo-projective. In this paper, we consider relationships between several relative injectivities and the invariance for certain homomorphisms in their injective hulls, and dually study several relative projectivities from the viewpoint of the dual invariant in their projective covers.…”

“…is the natural epimorphism ( [22]). Guil Asensio et al [9] called a dual automorphism-invariant module over a right perfect ring as automorphism coinvariant and proved that over a right perfect ring, a module M is automorphism coinvariant if and only if M is pseudo-projective. For the notion of automorphism invariant modules we refer to [7,23].…”

Indecomposable Non Uniserial Modules of Length Three

“…For more background on generalizations of projective modules we refer to [3] and [5]. For more background on quivers and their representations we refer to [1] and [2].…”

Модули, Коинвариантные Относительно Идемпотентных Эндоморфизмов Своих Накрытий

“…В [3] введено понятие псевдоинъективного модуля, т. е. модуля, в котором каждый мономорфизм из подмодуля модуля M в модуль M продолжается до некоторого эндоморфизма M. В [4] показано, что модуль M является псевдоинъективным в точности тогда, когда M автоморфизм-инвариантный модуль. Двойственное понятие к автоморфизм-инвариантным модулям под названием 1192 А. Н. Абызов, В. Т. Ле, К. К. Чюонг, А. А. Туганбаев автоморфизм-коинвариантные (или дуально автоморфизм-инвариантные) модули недавно изучено в [5][6][7].…”