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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
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