nilpotent element; nil clean element
Let $R$ be an associative unital ring and let $a\in R$ be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these results.
 Hirano, Y., Tominaga, H., Yaqub, A.: On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element
. Math. J. Okayama Univ. 30 (1988), 33-40. MR 0976729
| Zbl 0665.16016
 Šter, J.: Rings in which nilpotents form a subring
. Carpathian J. Math. 32 (2016), 251-258. MR 3587893