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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cyclic purity versus purity in excellent Noetherian rings
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by Melvin Hochster PDF
Trans. Amer. Math. Soc. 231 (1977), 463-488 Request permission


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 $P \in {\operatorname {Ass}}(A)$ 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 $S[[T]]$.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 231 (1977), 463-488
  • MSC: Primary 13D99
  • DOI:
  • MathSciNet review: 0463152