## 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

### Keywords:

multiplication ideal; Ohm type properties
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