Turkish Journal of Mathematics




We call a ring $R$ generalized right $\pi$-Baer, if for any projection invariant left ideal $Y$ of $R$, the right annihilator of $Y^n$ is generated, as a right ideal, by an idempotent, for some positive integer $n$, depending on $Y$. In this paper, we investigate connections between the \g $\pi$-Baer rings and related classes of rings (e.g., $\pi$-Baer, generalized Baer, generalized quasi-Baer, etc.) In fact, generalized right $\pi$-Baer rings are special cases of generalized right quasi-Baer rings and also are a generalization of $\pi$-Baer and generalized right Baer rings. The behavior of the generalized right $\pi$-Baer condition is investigated with respect to various constructions and extensions. For example, the trivial extension of the generalized right $\pi$-Baer ring and the full matrix ring over a generalized right $\pi$-Baer ring are characterized. Also, we show that this notion is well-behaved with respect to certain triangular matrix extensions. In contrast to generalized right Baer rings, it is shown that the generalized right $\pi$-Baer condition is preserved by various polynomial extensions without any additional requirements. Examples are provided to illustrate and delimit our~results.

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