| Summary: | <p> Generalized Matrix rings are ubiquitous in algebra and have relevant applications to analysis. A ring is <i>quasi-Baer</i> (<i> right p.q.-Baer</i>) in case the right annihilator of any ideal (resp. principal ideal) is generated by an idempotent. A ring is called <i> biregular</i> if every principal right ideal is generated by a central idempotent. A ring is called <i>right FI-extending</i> (<i> right strongly FI-extending</i>) if every fully invariant submodule is essential in a direct summand (resp. fully invariant direct summand). In this paper we identify the ideals and principal ideals, the annihilators of ideals and the central, semi-central and general idempotents of a 2 × 2 matrix ring. We characterize the generalized matrix rings that are quasi-Baer, p.q.-Baer and biregular and we present structural features of right FI-extending and right strongly FI-extending rings. We provide examples to illustrate these concepts.</p><p>
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