Prime ideals corresponding to primary ideals in decompositon

Question

Let $R$ be a commutative ring with unity different from $0$, and $I$ be a decomposable ideal of $R$, i.e. $$I=\mathfrak{q}_1\cap \mathfrak{q}_2 \cap \cdots \cap \mathfrak{q}_r $$ where (1) $\mathfrak{q}_i$ are primary ideals; (2) $rad(\mathfrak{q}_i)$ are all distinct; (3) No $\mathfrak{q}_i$ contains intersection of other $\mathfrak{q}_j$'s.

In this situation, it is well-known that

(1) $\mathfrak{p}_i$ are uniquely determined (independent of whether $R$ is noetherian or not).

(2) $\mathfrak{p}_i$ are precisely the prime ideals appearing in the set $\{ rad(I:x) \,\, | \,\, x\in R\}$.

If $R$ is noetherian, we can say more (see Prop.2.10(iii), page 21 for proof):

(2') $\mathfrak{p}_i$ are precisely the prime ideals appearing in the set $\{ (I:x) \,\, | \,\, x\in R\}$.

Q. Can't we conclude (2') if $R$ is not Noetherian?

Answer 1

Nice question!

I'm following the suggestion made by John Brevik in a comment.

The answer is No.

Here is a ring $R$ which does not satisfy (2'):

Let $K$ be a field, let $X_1,X_2,\dots$ be indeterminates, and set $$ R:=K[X_1,X_2,X_3,\dots]/(X_1,X_2^2,X_3^3,\dots). $$
Let $x_i$ be the image of $X_i$ in $R$. Clearly $\mathfrak m:=(x_1,x_2,\dots)$ is a maximal ideal of $R$, and is its nilradical. In particular the zero ideal of $R$ is $\mathfrak m$-primary, and it suffices to show that $\mathfrak m$ is not the annihilator of $r$ for any $r$ in $R$.

Let $B$ be the set of those monomials in $x_1,x_2,\dots$ which are of degree less than $i$ in $x_i$ for all $i$. I leave it to the reader to check that $B$ is a $K$-basis of $R$.

Let $r$ be a nonzero element of $R$. It suffices to show that we have $rx_i\ne0$ for some $i$.

If $i\ge2$ is such that $x_i$ does not occur in $r$ when $r$ is written as a $K$-linear combination of elements of $B$, then we have $rx_i\ne0$ as desired.