Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m,n$ fixed positive integers. (i) If ${\left(d[x,y]\right)}^m={[x,y]}_n$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathrm{Char}R\ne 2$ and ${[d\left(x\right),d\left(y\right)]}_m={[x,y]}^n$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n\left(I\right)$, there exist left invertible $U_1,U_2\in M_n\left(R\right)$ and right invertible $V_1,V_2\in M_n\left(R\right)$ such that $U_1{V}_{1}A{U}_2{V}_2=diag(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.

We characterize left Noetherian rings which have only trivial derivations.

An attractive interplay between the direct decompositions and the explicit form of basic subgroups in group rings of abelian groups over a commutative unitary ring are established. In particular, as a consequence, we give a simpler confirmation of a more general version of our recent result in this aspect published in Czechoslovak Math. J. (2006).