We define the class of UNI rings and present the results of their comprehensive investigations in connection with clean
rings, namely, our main result describes commutative UNI clean rings up to an isomorphism. This new concept is a
common generalization of the so-called UU rings examined by Danchev – Lam in (Publ. Math. Debrecen, 2016) and of
the so-called WUU rings studied by the author in (Tsukuba J. Math., 2016).
