examples of radicals of ideals in commutative rings

Let R be a commutative ring. Recall, that ideals I,J in R are called coprimeMathworldPlanetmathPlanetmath iff I+J=R. It can be shown, that if I,J are coprime, then IJ=IJ. Elements x1,,xnR are called pairwise coprime iff (xi)+(xj)=R for ij. It follows by induction, that for pairwise coprime x1,,xnR we have (x1xn)=(x1)(xn),

Let xR be such that


for some prime elementsMathworldPlanetmath piR, αi and assume that p1,,pn are coprime. Denote by


We shall denote by r(I) the radicalPlanetmathPlanetmathPlanetmath of an ideal IR.

Proposition. r((x))=(x¯).

Proof. ,,” Let α=max(α1,,αn). Then we have


and thus x¯α(x). This shows the first inclusion.

,,” Assume that yr((x)) and y0. Then there is n such that yn(x). Thus x divides yn. Of course for any i{1,,n} we have that pi divides x. Thus pi divides yn and since pi is prime, we obtain that pi divides y. Now for ij elements pi and pj are coprime, thus x¯ divides y and therefore y(x¯), which completesPlanetmathPlanetmath the proof.

Remark. If we assume that R is a PID (and thus UFD), then the previous proposition gives us the full characterization of radicals of ideals in R. In particular an ideal in PID is radical if and only if it is generated by an element of the form p1pn, where for ij elements pi and pj are not associated primes.

Examples. Consider ring of integersMathworldPlanetmath . Then we have:

Title examples of radicals of ideals in commutative rings
Canonical name ExamplesOfRadicalsOfIdealsInCommutativeRings
Date of creation 2013-03-22 19:04:34
Last modified on 2013-03-22 19:04:34
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Example
Classification msc 16N40
Classification msc 14A05
Classification msc 13-00