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Let $R$ be a commutative ring and $S\subseteq R$ a given multiplicative set. Let $(M,\le )$ be a strictly ordered monoid satisfying the condition that $0\le m$ for every $m\in M$. Then it is shown, under some additional conditions, that the generalized power series ring $\left[\left[R^{M,\le }\right]\right]$ is $S$-Noetherian if and only if $R$ is $S$-Noetherian and $M$ is finitely generated.

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 the ring...

In this paper, we study the existence of the $n$-flat preenvelope and the $n$-FP-injective cover. We also characterize $n$-coherent rings in terms of the $n$-FP-injective and $n$-flat modules.