On injective and surjective endomorphisms of finitely generated modules

Continuous modules are clean

“…For instance, if R is a regular ring with primitive factor rings Artinian, then every finitely generated module M R is a Fitting module [AFS,Theorem 2.5], and hence M R is a strongly clean module. In particular, the ring R itself is strongly π -regular and hence strongly clean.…”

Surjunctivity and Reversibility of Cellular Automata over Concrete Categories

“…8) In [35], it is shown that, for a commutative ring R, the category R-Mod f −g is surjunctive if and only if all prime ideals in R are maximal (if R is a nonzero ring, this amounts to saying that R has Krull dimension 0). The non-commutative rings R such that R-Mod f −g is surjunctive are characterized in [2]. 9) Let K be a field.…”

Construction of Modules With a Prescribed Direct Sum Decomposition

“…Hence, by (1), M is a direct sum of modules with local endomorphism ring. Also from (1) it follows that End R (M ) is a product of semiperfect rings.…”

“…Armendariz, Fisher and Snider were studying in [1] when every injective/onto endomorphism of a finitely generated module over a PI ring is bijective. They constructed in [1,Example 3.2] an example that was quite interesting for their context, but having a closer look people realized that their idea was giving a method to construct cyclic modules with a prescribed endomorphism ring. The further developments of Armendariz, Fisher and Snider's method have had an impact in the theory of direct sum decomposition of modules in general, and of direct sum decompositions of artinian modules in particular.…”