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Let R be a ring (not necessarily with identity) and let E denote the set of idempotents of R. We prove that R is a direct sum of a J-ring (every element is a power of itself) and a nil ring if and only if R is strongly π-regular and E is contained in some J-ideal of R. As a direct consequence of this result, the main theorem of [1] follows

international journal of mathematics and mathematical sciences

Rings decomposed into direct sums of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$J$"><mml:mi>J</mml:mi></mml:math>-rings and nil rings

Rings decomposed into direct sums ofJ-rings and nil rings