Isomorphisms Between Endomorphism Rings of Modules

Abstract

Abstract.We derive a necessary and sufficient condition that an isomorphism between the endomorphism rings of modules be induced by a semi-linear isomorphism of the modules. We require that at least one of the modules contain a non-zero free summand, generalizing work of Xu, who required both modules to be free.

A Baer-Kaplansky theorem for modules over principal ideal domains

“…Since B ∼ = R, it is not hard to see, using the same technique as in the proof for the bounded case of [6, Theorem 108.1], that ϕ is induced by an isomorphism ψ : R⊕G → R⊕H. Therefore the above theorem can be viewed as an improvement of [19,Theorem 2.1] for the case of principal ideal domains. Remark 4.…”

The Baer-Kaplansky theorem for all abelian groups and modules

“…The existence of an analogous statement for other kinds of modules (or abelian groups) was investigated by many authors, e.g. in [12], [15], [18], and [19]. There are situations in which there exist isomorphisms between endomorphism rings that are not induced in the above mentioned way.…”

“…There are situations in which there exist isomorphisms between endomorphism rings that are not induced in the above mentioned way. Such an example is described in [15, p. 486], and some other detailed studies are presented in [12] and [19]. On the other hand, there are examples which show that, in general, in order to obtain a Baer-Kaplansky Theorem, one needs to restrict to some reasonable classes of objects, see e.g.…”