Abstract
A nonzero element a in a unital ring R is called unit fine if there is a unit u in R such that ua is fine (i.e. a sum of a unit and a nilpotent). A ring all whose elements are unit fine is called accordingly. This turns out to be a new class of simple rings, including the class of fine rings (and so the class of simple Artinian rings). The paper studies unit fine elements and rings. For rings such that 1 is a sum of two units, it is proved that matrix rings over unit fine rings, are unit fine.
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