On Ideals of Regular Rings

摘要

In this paper, we investigate ideals of regular rings and give several characterizations for an ideal to satisfy the comparability. In addition, it is shown that, if I is a minimal two-sided ideal of a regular ring R, then I satisfies the comparability if and only if I is separative. Furthermore, we prove that, for ideals with stable range one, Roth's problem has an affirmative solution. These extend the corresponding results on unit-regularity and one-sided unit-regularity.