Annihilators and extensions of idempotent-generated ideals

Abstract

We define a ring R to be right -Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. This class of rings generalizes the class of right p.q.-Baer rings. We also define a ring R to be right I-extending if each ideal generated by an idempotent is essential, as a right R-module, in a right ideal generated by an idempotent. It is shown that the two conditions are equivalent in a semiprime ring. From the definition of a right -Baer ring, we define a generalization of a prime ring which is equivalent to the condition that all nonzero cyclic projective right R-modules are faithful. Such a ring is called right I-prime. For a semiprime ring, we show the existence of a -Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the -Baer hull. Polynomial and formal power series rings are studied with respect to the right -Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right -Baer then the formal power series ring over one indeterminate is right -Baer. The last section is devoted to matrix extensions of right -Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right -Baer is given. The last major theorem is a decomposition of a -Baer ring R, satisfying a mild finiteness condition, into a direct sum of rings where A is a direct sum of right I-prime rings; and B is a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.