A Note on Regularity Properties with Respect to Ideals

Abstract

The object of this article is to study the regularity properties of elements of a ring with respect to a given ideal I. As expected, several concepts that are equivalent in the case of I = R turn out to be distinct for a general ideal I and we consider the relations between these properties. In particular, we replace the set of units U(R) of the ring R by the set U _I (R) = {u|uI = Iu = I} and use these “relative units” to obtain generalizations of notions such as stable range and unit-regularity. We also see that on assuming the set of “relative units” to have no zero divisors, we can obtain several interesting results.

2000 Mathematics Subject Classification:

• 16E50

ACKNOWLEDGMENTS

The author is grateful to the referee for useful comments and a very careful reading of the article. In particular, the references Chen (2004a, 2006) were suggested by the referee.

Notes

Communicated by I. Swanson.