This example appears as Example 1 in the following reference:

Melvin Hochster, *Non-openness of loci in Noetherian rings,* Duke Math. J. **40** (1973), 215–219. MR311653. ZBL0257.13015. DOI: 10.1215/S0012-7094-73-04020-9.

In fact, the example in question is a special case of the general result quoted below.

[…] $K$ denotes a field and all otherwise unspecified tensor products
are taken over $K$. If $R$ is a $K$-algebra, we say that $R$ is
*absolutely* Noetherian over $K$ if for every overfield $L \supset K$, $L \otimes R$ is Noetherian. If $R$ is a localization of a finitely
generated $K$-algebra, then $R$ is absolutely Noetherian over $K$. $R$
is *absolutely* a domain over $K$ if, likewise, each $L \otimes R$ is
a domain, and $P$ is absolutely prime if, equivalently, either $R/P$
is absolutely a domain or each $L \otimes P$ is prime (in $L \otimes R$).

[…]

**Proposition 1.** *Let $\{R_i\}_{i \in I}$ be a family of absolutely Noetherian $K$-algebras indexed by an
infinite set $I$, and for each $i \in I$, let $P_i$ be a nonzero
absolutely prime ideal of $R_i$. Assume that each $R_i$ is absolutely
a domain.*

*Let $R' = \bigotimes_{i \in I} R_i$. Then $R'$ is a domain and for
each $i$, $P_iR'$ is prime. Moreover, if $S = R' - (\bigcup_i P_iR')$
and $R = S^{-1}R'$, then $R$ is a Noetherian domain whose maximal
ideals are in one-to-one correspondence with $I$ via the map $i \mapsto P_iR$. In addition, each nonzero element of $R$ belongs to
only finitely many maximal ideals, and for any maximal ideal $P_iR$ of
$R$* $$R_{P_iR} \cong (L_i \otimes R_i)_{P_i^e},$$ *where $L_i$ is a
certain extension field of $K$ and $P_i^e$ is $P_i(L_i \otimes R_i)$.*

*In particular, if for each $i \in I$, $R_i$ is a subring of a
polynomial ring over $K$ and is generated by a finite nonempty set of
forms of positive degree and $P_i$ is the ideal generated by these
forms, then the hypotheses of the first paragraph are satisfied, and
the conclusions of the second paragraph hold. Moreover, the local
rings of $R$ are algebro-geometric in this case.*

In Example 1, Hochster applies the Proposition to the situation where $I$ is the set of positive integers and $K$ is an arbitrary field, in which case he sets $R_i = K[x_i^2,x_i^3]$ and $P_i = (x_i^2,x_i^3)R_i$ for each $i$. Then, $R_{P_iR} \cong L_i[x_i^2,x_i^3]_{P_i^e}$ for every $i$, which is a non-normal local domain of dimension one. By the Proposition these are the localizations of $R$ at nonzero prime ideals, and the only regular point in $\operatorname{Spec}R$ is the generic point corresponding to the zero ideal.

It is useful to note the following consequence of Proposition 1, also in Hochster's paper:

**Proposition 2.** *Let $\mathscr{P}$ be a property of local rings and suppose there is an algebro-geometric local ring $(R_1,P_1)$ over a
field $K$ such that*

*$R_1$ is absolutely a domain,*
*$P_1$ is absolutely prime, and*
*for every overfield $L \supset K$* $$(L \otimes R_1)_{P_1^e}$$ *fails to have property $\mathscr{P}$.*

*Suppose that each field $L \supset K$ has $\mathscr{P}$. Then there is a locally algebro-geometric Noetherian domain $R$ over $K$ in which the $\mathscr{P}$ locus is not open.*

This gives a nice way to construct rings that are locally excellent but not excellent, for instance.