A Note on Regularity Properties with Respect to Ideals

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.