Further results on skew monoid rings of a certain free monoid (I)

Let R be a ring with an endomorphism $\sigma$, F be a free monoid and S be a factor of $F\cup \{0\}$ such that $(S\setminus \{1\})^{n}=0$ for some positive integer $n \ge 2$. The second author and Moussavi [Annihilator properties of skew monoid rings, Comm. Algebra, 42 (2) (2014), 842–852] started studying the skew monoid ring $R[S;\sigma ]$. In this paper, we continue the study of these rings