Abstract
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.
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