A Generalization of $\Delta$U Rings

In this paper, we introduce and study a new class of rings calling them {\it weakly $\Delta U$-rings}, hereafter abbreviated as {\it $W\Delta U$-rings} for short. A ring $R$ is said to be $W\Delta U$ if every unit of $R$ can be expressed as $\pm 1 + d$ for some $d \in \Delta(R)$, where $\Delta(R)$ is the largest Jacobson radical of $R$ that is closed under multiplication by units. Utilizing the known structure of $\Delta(R)$, we investigate the relationships between $W\Delta U$ rings and certain classical concepts such as $\Delta U$-rings, $UJ$-rings, $WUJ$-rings, as well as clean and exchange rings. Among the main results, we show that a matrix ring $M_n(R)$ is never $W\Delta U$ for any $n \ge 2$. We also provide complete characterizations of local, semi-local, semi-simple and semi-regular rings that are $W\Delta U$. Furthermore, it is shown for exchange rings that the $W\Delta U$ property is equivalent to being $WUJ$. Furthermore, the behavior of $W\Delta U$-rings under various ring extensions, including skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions and group rings, is thoroughly examined. Several examples are given to illustrate that the class of $W\Delta U$-rings properly contains the class of $\Delta U$-rings. Finally, necessary and sufficient conditions for a group ring $RG$ to be $W\Delta U$ are established too. Resuming all of the presented above, our results expanded those by Karaba\c{c}ak et al. published in J. Algebra \& Appl. (2021).