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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As a special case of fully invariant subgroups, strongly invariant subgroups are introduced and studied for Abelian groups.

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While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.

A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly the class of all simple artinian rings. One of the main results in this paper is that matrix rings over fine rings are always fine rings. This implies, in particular, that any nonzero (square) matrix over a division ring is the sum of an invertible matrix and a nilpotent matrix.

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In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice G] is the union of these intervals.2000 Mathematics subject classification: primary 06C99, 20K10, 20K27.