Linearly equivalent topologies and locally quasi-unmixed rings

Abstract

Let $\bar{I}$ denote the integral closure of an ideal in a  Noetherian ring $R$. The main result of this paper asserts that $R$  is locally quasi-unmixed if and only if, the topologies defined by $\overline{I^n}$  and $I^{\langle n\rangle}$, $\ n\geq 1$,  are equivalent. In addition, some results about the behavior of linearly equivalent  topologies of ideals under various ring homomorphisms are included.

Keywords

• associated primes,linearly equivalent topologies,integral closure,locally quasi-unmixed ring,Rees ring

References

1. [1] H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.
2. [2] S. McAdam, Asymptotic Prime Divisors, Lecture Notes in Math. 1023, Springer- Verlag, New York, 1983.
3. [3] S. McAdam, Quintasymptotic primes and four results of Schenzel, J. Pure Appl. Algebra 47, 283–298, 1987.
4. [4] S. McAdam and L. J. Ratliff Jr., On the asymptotic cograde of an ideal, J. Algebra 87, 36–52, 1984.
5. [5] M. Nagata, Local Rings, Interscience, New York, 1961.
6. [6] R. Naghipour, Locally unmixed modules and ideal topologes, J. Algebra 236, 768–777, 2001.
7. [7] D.G. Northcott and D. Rees,Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50, 145–158, 1954.
8. [8] L.J. Ratliff Jr., Asymptotic sequences, J. Algebra 85, 337–360, 1983.

9. [9] L.J. Ratliff Jr., On asymptotic prime divisors, Pacific J. Math. 111, 395–413, 1984.
10. [10] L.J. Ratliff Jr., Asymptotic prime divisors and integral extension rings, J. Algebra 95, 409–431, 1985.
11. [11] D. Rees, A note on analytically unramified local rings, J. London Math. Soc. 36, 24–28, 1961.
12. [12] J.K. Verma, On ideals whose adic and symbolic topologies are linearly equivalent, J. Pure Appl. Algebra 47, 205–212, 1987.