Cyclic purity versus purity in excellent Noetherian rings

A characterization is given of those Noetherian rings R such that whenever R is ideally closed ($\equiv$ cyclically pure) in an extension algebra S, then R is pure in S. In fact, R has this property if and only if the completion $(A,m)$ of each local ring of R at a maximal ideal has the following two equivalent properties:

(i) For each integer $N > 0$ there is an m-primary irreducible ideal ${I_N} \subset {m^N}$.

(ii) Either $\dim \;A = 0$ and A is Gorenstein or else depth $A \geqslant 1$ and there is no $P \in {\text{Ass}}(A)$ such that $\dim (A/P) = 1$ and $(A/P) \oplus (A/P)$ is embeddable in A.

It is then shown that if R is a locally excellent Noetherian ring such that either R is reduced (or, more generally, such that R is generically Gorenstein), or such that Ass(R) contains no primes of coheight $\leqslant 1$ in a maximal ideal, and R is ideally closed in S, then R is pure in S. Matlis duality and the theory of canonical modules are utilized. Module-theoretic analogues of condition (i) above are, of necessity, also analyzed.

Numerous related questions are studied. In the non-Noetherian case, an example is given of a ring extension $R \to S$ such that R is pure in S but $R[[T]]$ is not even cyclically pure in $S[[T]]$.