Danchev P. V.

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).

We prove some properties of $n$-totally projective abelian $p$-groups. Under some additional conditions for the group structure, we obtain an equivalence between the notions of $n$-total projectivity and strong $n$-total projectivity. We also show that $n$-totally projective $A$-groups are isomorphic if they have isometric $p^n$-socles.