Rings with kernel inclusion quasiorder

Abstract

Building on previous work for semigroups of functions and binary relations, we axiomatize structures consisting of endomorphisms of abelian groups equipped with composition, the usual pointwise operations, and the quasi-order of kernel inclusion. The resulting structures are associative rings enriched by a quasiorder satisfying a finite set of laws. More generally, we axiomatize the kernel inclusion quasi-order on the ring R induced by a right R-module, and we call the resulting abstract structures rings with ker-order. A characterisation of the possible ker-orders on a fixed ring is given in terms of certain families of its right ideals. The lattice of all ker-orders on the ring of rational integers is described.