Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$ generated by $a$ and $b$ is contained in the union $\bigcup_{i\in I}\mathfrak{p}_i$, can we deduce that $\langle a, b\rangle \subseteq \mathfrak{p}_j $ for some $j\in I$?

## 1 Answer

This is an interesting question. Quentel's Example provides an example. Let $R$ be a reduced ring, $Min(R)$ its space of minimal prime ideals, and $q(R)$ its classical quotient ring.

**Theorem** The following are equivalent.

$q(R)$ is

*von Neumann regular*.$R$ satisfies Property $A$ and $Min(R)$ is compact.

$Min (R)$ is compact and whenever a f.g. ideal is contained in the union of minimal prime ideals, then it is contained in one them.

Consequently, Quentel's Example is a reduced ring $R$ with $Min(R)$ compact but $q(R)=R$ not von Neumann regular. Therefore, it must possess a f.g. ideal which is contained in a union of minimal prime ideals but contained in any single one.

Glaz, Sarah
*Controlling the zero divisors of a commutative ring.* Commutative ring theory and applications (Fez, 2001), 191–212,
Lecture Notes in Pure and Appl. Math., 231, Dekker, New York, 2003.

J. A. Huckaba, Commutative Rings With Zero Divisors, Marcel Dekker, Inc. (1988)

finiteunion of prime ideals is contained in one of those prime ideals. In particular, since a noetherian commutative ring has finitely manyminimalprime ideals, the answer is positive then, so possible counterexamples are to be looked for outside the noetherian world. $\endgroup$