Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma$ be an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. Then $R$ is said to be an almost $\delta$-divided ring if every minimal prime ideal of $R$ is $\delta$-divided. Let $R$ be a Noetherian ring which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers). Let $\sigma$ be an automorphism of $R$ such that $R$ is a $\sigma (*)$-ring and $\delta$ a $\sigma$-derivation of $R$ such that $\sigma \left(\delta \right(a\left)\right)=\delta \left(\sigma \right(a\left)\right)$ for all $a\in R$. Further, if for any...

It is well known that the monoid ring of the free product of a free group and a free monoid over a skew field is a fir. We give a proof of this fact that is more direct than the proof in the literature.