Valuation domains have been
extended in the non-commutative case by several authors, giving rise to the so called
generalized valuation rings, that is, rings whose lattice of right ideals is linearly
ordered by inclusion. We propose here the study of rings whose lattice of kernel
functors is linearly ordered and we indicate throughout this article similarities
between them and valuation rings. In addition, the rings presented here include
generalized valuation rings and coincide with them when commutativity is
assumed. They therefore provide a new non-commutative analogue of valuation
rings.
Unlike the generalized valuation rings, the rings we study enjoy properties that
transfer nicely to matrix rings thus enabling us to treat questions in a broader
context.
Finally, a semigroup structure imposed on the lattice of kernel functors is
analyzed and the article concludes by examining when that semigroup can be
thought of as the semigroup of a valuation ring.
