Generalized Depth and Associated Primes in the Perfect Closure $R^\infty$

Abstract

\indent Letting $(S, \mathfrak{n})$ be a Noetherian local ring, and $M$ be a finitely generated $S$-module, the notions of $\depth_S(M)$ and associated primes over $M$, denoted $\Ass_S(M)$, are fundamental concepts in commutative algebra. However, if $S$ is non-Noetherian, both of these notions become more subtle. Prime ideals in this scenario may then be categorized as associated primes, weakly associated primes, strong Krull primes, and Krull primes, respectively $\Ass_S(M)$, $\wAss_{S} (M)$, $\sK_S(M)$, and $\K_S(M)$. Likewise, any study of depth must distinguish between $\cdepth_S(M)$, $\kdepth_S(M)$, and $\rdepth_S(M)$.