This paper deals with the rings which satisfy $DC{C}_d$ condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of rings...

Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ such that either $x^p[x^n,y]x^q=x^r[x,y^m]y^s$ or $x^p[x^n,y]x^q=y^s[x,y^m]x^r$ for all $y\in R$. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q\left(m\right)$ (i.e. for all $x,y\in R,m[x,y]=0$ implies $[x,y]=0$).

A ring $R$ is called a left APP-ring if the left annihilator $l_R\left(Ra\right)$ is right $s$-unital as an ideal of $R$ for any element $a\in R$. We consider left APP-property of the skew formal power series ring $R\left[\right[x;\alpha \left]\right]$ where $\alpha$ is a ring automorphism of $R$. It is shown that if $R$ is a ring satisfying descending chain condition on right annihilators then $R\left[\right[x;\alpha \left]\right]$ is left APP if and only if for any sequence $(b_0,b_1,\cdots )$ of elements of $R$ the ideal $l_R$ $\left({\sum }_{j=0}^{\infty }{\sum }_{k=0}^{\infty }R{\alpha }^k\left(b_j\right)\right)$ is right $s$-unital. As an application we give a sufficient condition under which...