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For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper wediscuss an extension of $R$ called its perfect closure $R^\infty$. Thisextension contains all $p^e$-th roots of elements of $R$, and is usuallynon-Noetherian. We first define the generalized notions of associated primes ofa module over a non-Noetherian ring. Then for any $R$-module $M$, we state acorrespondence between certain generalized prime ideals of $(R^\infty \otimes_RM)/N$ over $R^\infty$, and the union of associated prime ideals of $F^e(M)/N_e$as $e \in \mathbb{N}$ varies. Here $F$ refers to the Frobenius functor, and inthe paper we define an $F$-sequence of submodules $\lbrace N_e \rbrace\subseteq \lbrace F^e(M) \rbrace$ as $e$ varies, while $\underrightarrow{\lim}\ N_e = N$. Under the further assumptions that $M$ is finitely generated and$(R,\mathfrak{m})$ is an $F$-pure local ring, we then show thatdepth$_R(F^e(M))$ is constant for $e \gg 0$, and we call this value thestabilizing depth, or s depth$_R(M)$. Lastly, we turn to non-Noetherianmeasures of the depth of $R^\infty \otimes_R M$ over $R^\infty$, whichgeneralize as well. Two of these values are the k depth and the c depth, and weshow k depth$_{R^\infty} (R^\infty \otimes_R M) =$ s depth$_R (M) \geq$ cdepth$_{R^\infty} (R^\infty \otimes_R M)$, while all three values are equalunder certain assumptions.

Generalized depth and associated primes in the perfect closure $R^\infty$