ON REGULAR SUBDIRECT PRODUCTS OF SIMPLE ARTINIAN-RINGS

We construct a counterexample to settle simultaneously the following questions all in the negative: (1) Is a regular subdirect product of simple artinian rings unit-regular? (2) If R is a regular ring such that every nonzero ideal of R contains a nonzero ideal of bounded index, is R unit-regular? (3) Is a regular ring with a Hausdorff family of pseudo-rank functions unit-regular? (4) If R is a regular ring which contains no infinite direct sum of nonzero pairwise isomorphic right ideals, is R unit-regular? (5) Is a regular Schur ring unit-regular? © 1990 by Pacific Journal of Mathematics.