If R is a field and it has two ideal and any union of two ideal is again ideal. But Can we give an example of commutative ring(necessarily not field) with 1 and union of two ideal is again ideal?
2026-04-03 15:56:40.1775231800
Nontrivial example of a ring in which the union of ideals is an ideal
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Any uniserial ring (in the sense that the ideals are linearly ordered) has the property that unions of ideals are ideals.
So, for example, $F[[x]]$ for a field $F$, or $\mathbb Z/(p^n)$ for a prime $p$ and positive integer $n$, or any of these DaRT results.
(Among commutative rings) Uniserial rings are exactly the rings for which the union of any two ideals is an ideal. If $A$ and $B$ are two ideals, and $a\in A\setminus B$ and $b\in B\setminus A$, then it's easy to see $a+b\notin A\cup B$.