*π-dual Baer Modules and π-dual Baer Rings*

Derya Keskin Tütüncü & **Rachid Tribak **

Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey.

Centre Régional des Métiers de l’Education et de la Formation (CRMEF-TTH)-Tanger,

Avenue My Abdelaziz, Souani, B.P. 3117, Tangier, Morocco

Pages 108-123 | Received 08 November 2022, Accepted 20 February 2023, Published 16 June 2023

**Abstract**

Let R be a ring and let M be an R-module with \mathbf{S}=End_R(M). A submodule N of M is said to be *projection invariant *in M (denoted N \unlhd_p M) if eN \subseteq N for all e=e^2 \in \mathbf{S}. We call M \pi–*dual Baer,* if for each N \unlhd_p M there exists e^2=e\in \mathbf{S} such that {f\in \mathbf{S}\mid f(M)\subseteq N}= e\mathbf{S}. A characterization of \pi-dual Baer modules is provided. We show that the class of \pi-dual Baer modules lies strictly between the classes of dual Baer modules and quasi-dual Baer modules. It is also shown that in general, the class of \pi-dual Baer modules is neither closed under direct sums nor closed under direct summands. The structure of \pi-dual Baer modules over Dedekind domains is completely determined. We conclude the paper by studying right \pi-dual Baer rings. We call a ring R *right* \pi–*dual Baer* if the right R-module R_R is right \pi-dual Baer. A characterization of this class of rings is provided. We also investigate the transfer between a base ring R and many of its extensions (for example, full matrix rings over R or R[x] or R[[x]]). In addition, we characterize the 2-by-2 generalized triangular right \pi-dual Baer matrix rings.

**Keywords**: dual Baer module; quasi-dual Baer module; π-dual Baer module; endomorphism rings; projection invariant submodule.

**MSC numbers**: Primary 16D10, 16S50; Secondary 16D80.

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