Symmetrization in clean and nil-clean rings

We introduce and investigate D-clean and D-nil-clean rings as well as some other closely related symmetric versions of cleanness and nil-cleanness. A comprehensive structural characterization is given for these symmetrically clean and symmetrically nil-clean rings in terms of Jacobson radical and its quotient. It is proved that strongly clean (resp., strongly nil-clean) rings are always D-clean (resp., D-nil-clean).Our results corroborate our recent findings published in Bull. Irkutsk State Univ., Math. (2019) and Turk. J. Math. (2019). We also show that weakly nil-clean rings defined as in Danchev-McGovern (J. Algebra, 2015) and Breaz – Danchev – Zhou (J. Algebra & Appl., 2016) are actually weakly nil clean in the sense of Danchev-Šter (Taiwanese J. Math., 2015). This answers the question of the reviewer D. Khurana (Math. Review, 2017).