QUINTESSENTIAL PRIMES AND IDEAL TOPOLOGIES OVER A MODULE

Abstract

Let I be an ideal of a Noetherian ring R, N a finitely generated R-module and let S be a multiplicatively closed subset of R. We define the Nth (S)-symbolic power of I w.r.t. N as S(I ⁿ N) = ∪ _s∈S (I ⁿ N: _N s). The purpose of this paper is to show that the topologies defined by {Iⁿ N} _n≥0 and {S(Iⁿ N)} _n≥0 are equivalent (resp. linearly equivalent) if and only if S is disjoint from the quintessential (resp. essential) primes of I w.r.t. N.

ACKNOWLEDGMENTS

I am deeply grateful to thank Professor H. Zakeri for many helpful discussions, as well as Dr. K. Divaani-Aazar for offering many nice suggestions that improved this paper. Thanks are due to Professor L. J. Ratliff, Jr. for his useful comments, and to the referee for careful reading of the original manuscript and valuable suggestions. Finally, the author would like to thank from the University of Tabriz and Institute for Studies in Theoretical Physics and Mathematics for their financial supports.