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Radicals of PID's and Dedekind Domains

Published online by Cambridge University Press:  20 November 2018

R. E. Propes
R. E. Propes*
Affiliation:
The University of Wisconsin-Milwaukee, Milwaukee, Wisconsin
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The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, if

then T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 566 - 572
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Clark, A., Elements of abstract algebra (Wadsworth, Belmont, Calif., 1971).Google Scholar
2. Hoffman, A., Direct sum closure properties of radicals, J. Natur. Sci. and Math. 10 (1970), 53–58.Google Scholar
3. Kaplansky, I., Fields and rings (University of Chicago Press, Chicago, 1970).Google Scholar
4. Lee, Y. L., On the construction of lower radical properties, Pacific J. Math. 28 (1969), 393–395.Google Scholar
5. Lee, Y. L., On the construction of upper radical properties, Proc. Amer. Math. Soc. 19 (1968), 1165–1166.Google Scholar
6. Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (Van Nostrand, New York 1958).Google Scholar