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The strong $P$-cleanness over rings

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 $2\times 2$ matrices over commutative local rings are also determined.