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.