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Abstract. An element of a ring R is strongly P -clean provided that it can be written as the sum of an idempotent and a strongly nilpotent element that commute. A ring R is strongly P -clean in case each of its elements is strongly P -clean. We investigate, in this article, the necessary and sufficient conditions under which a ring R is strongly P -clean. Many characterizations of such rings are obtained. The criteria on strong P -cleanness of 2 × 2 matrices over commutative projective-free rings are also determined.Mathematics Subject Classification (2010): 16S50, 16U99.

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In this paper, we focus on the duo ring property via quasinilpotent elements, which gives a new kind of generalizations of commutativity. We call this kind of rings qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others, it is proved that if the Hurwitz series ring $H(R; \alpha)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1.

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