Abstract. Generalizing the notion of nil cleanness from [9], in parallel to [8], we define the concept of weak nil cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

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We characterize the nil clean matrix rings over fields. As a by product, we obtain a complete characterization of the finite rank Abelian groups with nil clean endomorphism ring and the Abelian groups with strongly nil clean endomorphism ring, respectively. * S. Breaz is supported by the CNCS-UEFISCDI grant PN-II-RU-TE-2011-3-0065. †

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Abstract. We define and completely describe the structure of invo-clean rings having identity. We show that these rings are clean but not (weakly) nil-clean and thus they possess independent properties than these obtained by Diesl in [7] and by Breaz-Danchev-Zhou in [2].