-closures of ideals and rings

Author:
Louis J. Ratliff

Journal:
Trans. Amer. Math. Soc. **313** (1989), 221-247

MSC:
Primary 13A15; Secondary 13B20, 13C99

DOI:
https://doi.org/10.1090/S0002-9947-1989-0961595-2

MathSciNet review:
961595

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if is a commutative ring with identity and is a multiplicatively closed set of finitely generated nonzero ideals of , then the operation is a closure operation on the set of ideals of that satisfies a partial cancellation law, and it is a prime operation if and only if is -closed. Also, if none of the ideals in is contained in a minimal prime ideal, then , the integral closure of in , and if is the set of all such finitely generated ideals and contains an ideal in , then . Further, has a natural -closure is a closure operation on a large set of rings that contain as a subring, behaves nicely under certain types of ring extension, and every integral extension overring of is for an appropriate set . Finally, if is Noetherian, then the associated primes of are also associated primes of and for all .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0961595-2

Keywords:
ACC on colon ideals,
cancellation law,
closure operation,
directed set,
flat extension ring,
integral closure of an ideal,
integral extension ring,
Noetherian ring,
polynomial extension ring,
prime divisor,
prime operation,
reduction of an ideal,
Rees ring,
semiprime operation

Article copyright:
© Copyright 1989
American Mathematical Society