S. M. N. Sheykholislami

arXiv2018

On radical and torsion theory in the category of S-acts

In abelian categories like the category of R-modules and even in the category S-Act 0 of S-acts with a unique zero, idempotent radicals and torsion theories are equivalent, and the τ-torsion and τ- torsion free classes of a torsion theory τ are closed under coproducts. These are not necessarily true in the category S-Act of S-acts. In this paper, we prove that torsion theories are equivalent with the Kurosh- Amitsur radicals. We, also, show that the class of Kurosh-Amitsur radicals is a reflective subcategory of Hoehnke radicals, as a poset.

math.RT↗

arXiv2018

Radical-injectivy in the category S-Act

Various generalizations of the concept of injectivy, in particular injectivy with respect to a specific class of morphisms, have been intensively studied throughout the years in different categories. One of the important kinds of injectivy studied in the category R-Mod of R-modules is $τ$-injectivy, for a torsion theory $τ$, or in the other words r-injectivy, where r is the induced idempotent radical by $τ$. In this paper, we introduce the notion of r-injectivy, for a Hoehnke radical r in the category S-Act of S-acts and we study the main properties of this kind of injectivy. Indeed, we show that this kind of injectivy is well behavior and also we present a Bear Theorem for r-injective S-acts. We then consider r-injectivy for a Kurosh-Amitsur radical r and we give stronger results in this case. Finally we present conditions under which r-injective S-acts are exactly injective ones and we give a characterization for injective S-acts

math.RT↗