Rings with uniformly $S$-$SFT$

Abstract

In this article, we examine the notion of uniformly $S$-SFT and
study its properties. Let $R$ be a commutative ring and $S$ a
multiplicative subset of $R.$ A ring $R$ is said to be uniformly
$S$-SFT if there exists an element $s$ in $S$ such that for every
ideal $I$ of $R$, there exist a finitely generated sub-ideal $J$ of
$I$ and a positive integer $n$ with the property that $sa^n\in J $
for all $a$ in $I$. Our investigation includes proving Cohen's
Theorem for uniformly $S$-SFT rings and analyzing the behavior of
uniformly $S$-SFT sets under various ring operations like Nagata's
idealization and amalgamation of algebras.