## A note on unions of ideals and cosets of ideals.(English)Zbl 0827.13001

Let $$A$$ be a commutative ring with 1, let $$M$$ be an arbitrary $$A$$-module. Let $$m\text{-spec} (A)$$ be the set of all maximal ideals of $$A$$, $$m \text{-supp} (M) = m\text{-spec} (A) \bigcap \text{supp} (M)$$ and $$N$$ be the set of all natural numbers. With this notation, the author establishes the following results:
Result 1. Let $$I$$, $$I_1, \ldots, I_r$$ be ideals, $$I_{r + 1}, \ldots, I_{r + s}$$ prime ideals of $$A$$, where $$r,s \in N$$, $$r + s \geq 1$$. If $$r \leq \min \{\text{card} (A/{\mathfrak m}:{\mathfrak m}\in m\text{-spec} (A)\}$$, then $$I \subset \bigcup^{r + s}_{i = 1} I_i$$ implies $$I \subset I_i$$ for some $$i$$ with $$1 \leq i \leq r + s$$.
Result 2. Let $$I$$, $$I_1, \ldots, I_t$$ be ideals of $$A$$, $$t \in N$$, $$a_{ij} \in A$$, $$j = 1, \ldots, n_i$$, $$n_i \in N$$ for $$i = 1, \ldots, t$$. Suppose $$I \not \subset I_i$$ for all $$i$$ and $$\sum^t_{i = 1} n_i < \inf \{\text{card} (A/{\mathfrak m}) :{\mathfrak m}\in m\text{-spec} (A)\}$$. Then $$I \not \subset \bigcup^t_{i = 1} \bigcup^{n_i}_{j = 1} (a_{ij} + I_i)$$.

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings

### Keywords:

prime ideals; maximal ideals
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