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The Ohm type properties for multiplication ideals. (English) Zbl 1054.13500

Let \(R\) be a commutative ring with identity. Given a nonempty collection \(\{I_\lambda \}_{\lambda \in \Lambda }\) of ideals with \(I=\sum_\lambda I_\lambda \), it is of interest to know when
(\({*}k\)) \((\sum_\lambda I_\lambda )^k=\sum_\lambda I_\lambda ^k\), or
(\({*}{*}k\)) \((\bigcap_\lambda I_\lambda )^k=\bigcap_\lambda I_\lambda ^k\)
holds. For example, it is well known that if \(R\) is a Prüfer domain then (\({*}k\)) holds for all natural numbers \(k\) while (\({*}{*}k\)) holds for all natural numbers \(k\) when \(\Lambda \) is finite, but need not hold in general. The purpose of this paper is to determine more general conditions under which (\({*}k\)) or (\({*}{*}k\)) holds. The author first proves that if \(I=\sum_\lambda I_\lambda \) is a multiplication ideal (i.e., each ideal contained in \(I\) is a multiple of \(I\)), then (\({*}k\)) holds for each \(k\geq 1\).
He then claims that if \(I=I_1+\cdots +I_n\) is a multiplication ideal, then \((I_1\cap \cdots \cap I_n)^k=I_1^k\cap \cdots \cap I_n^k\). This is not correct as may be seen by taking \(R=K[X^2,X^3]\), \(K\) a field, \(I_1=X^2R\), \(I_2=X^4R\), \(I_3=X^5R\) and \(k=2\). (The proof, while correct for the case \(n=2\), seems to assume that a subsum of \(n-1\) ideals from \(\{I_1,\ldots ,I_n\}\) is a multiplication ideal.) This error aside, the paper does contain a number of other correct results involving multiplication ideals.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
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