Abstract
An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring [Formula: see text] is strongly unit nil-clean, if for any [Formula: see text] there exists a unit [Formula: see text], such that [Formula: see text] is strongly nil-clean. We prove, in this paper, that a ring [Formula: see text] is strongly unit nil-clean, if and only if every element in [Formula: see text] is equivalent to a strongly nil-clean element, if and only if for any [Formula: see text], there exists a unit [Formula: see text], such that [Formula: see text] is strongly [Formula: see text]-regular. Strongly unit nil-clean matrix rings are investigated as well.
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