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The aim of the paper is to characterize the equation Formula: see text, where Formula: see text that is the largest Jacobson radical subring of Formula: see text and Formula: see text is the set of invertible elements of a ring Formula: see text. We show that this equation is closely related to Formula: see text-rings and rings whose elements can be written as the sum of an idempotent and an element from Formula: see text. After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation Formula: see text within many well-studied classes of rings. Finally, we close the paper with group rings.
Karabacak et al. (Wed,) studied this question.
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