Perspectivity and cancellation in regular rings

Abstract

Let R be a (von Neumann) regular ring, and L(R) the compIemented modular lattice of principal right ideals of R. In [6], it is shown that for suitable regular rings, and suitable modules A, B, C, z4@BzA@C implies B N C. This is a type of cancellation law that works e.g., for finitely generated projective modules over a “finite” self-injective regular ring. It turns out that for finitely generated projective modules over regular rings this cancellation law holds precisely when the ring is unit regular [definitions follow). A related problem is the transitivity of perspectivity on L(R). We show that transitivity holds on a two by two matrix ring over R if and only if R is unit regular. An example of Bergman’s is included to show that transitivity on L(R) itself is not sufficient for unit regularity. Along the way, we obtain the following result for arbitrary complemented lattices: If perspectivity is cancellative (in the sense above), it is transitive; if perspectivity is transitive, it is additive (in the obvious sense). Another example, also due to Bergman, is included. I am also indebted to George Bergman for many ideas and results, in particular the concept of a von Neumann module. All rings are associative and possess I. A ring is regular if all finitely generated right (or left) modules are generated by an idempotent. Equivalently, every element a of R possesses a quasi-inverse; that is, an element h with aba = b. The collection of principal right ideals is then a complemented modular lattice, which we denote by L(R). We use either of two notations: eR (for e = es in R) or JEL(R) to denote a principal right ideal If