Asymmetry in the lattice of kernel functors

Abstract

Much of the research done by different authors on the lattice of kernel functors (equivalently, linear topologies) has been summarized by Golan in [2]. More recently, the rings whose lattices of kernel functors are linearly ordered were introduced in [3] as a categorical generalization of valuation rings in the non-commutative case. Results (and examples) in [3] show that there is an abundance of non-commutative rings R whose lattices (R), both in Mod-R and R-Mod, are simultaneously linearly ordered; however, the question of the symmetry of this condition remained open. Here we will prove that, for every natural number n≥3, there exists a ring Rn such that (Mod-Rn) is a linearly ordered lattice of n elements, whereas (Rn-Mod) is not linearly ordered.