Non-Commutative Residuated Lattices

Abstract

In the theory of non-commutative rings certain distinguished subrings, one-sided and two-sided ideals, play the important roles. Ideals combine under crosscut, union and multiplication and hence are an instance of a lattice over which a non-commutative multiplication is defined.† The investigation of such lattices was begun by W. Krull (Krull [3]) who discussed decomposition into isolated component ideals. Our aim in this paper differs from that of Krull in that we shall be particularly interested in the lattice structure of these domains although certain related arithmetical questions are discussed.

Presented to the Society in two parts: April 9, 1938, under the title Non-commutative residuation, and November 26, 1938, under the title Archimedian residuated lattices; received by the editors May 1, 1939.