Strongly nilclean corner rings

We show that if R is a ring with an arbitrary idempotent e such that eRe and (1 − e)R(1 − e) are both strongly nil-clean rings, then R/J (R) is nil-clean. In particular, under certain additional circumstances, R is also nil-clean. These results somewhat improves on achievements due to Diesl in J. Algebra (2013) and to Ko¸san-Wang-Zhou in J. Pure Appl. Algebra (2016). In addition, we also give a new transparent proof of the main result of Breaz-Calugareanu-Danchev-Micu in Linear Algebra Appl. (2013) which says that if R is a commutative nil-clean ring, then the full n × n matrix ring Mn(R) is nil-clean.