Continuous closure, axes closure, and natural closure

Let $R$ be a reduced affine $\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\mathbb C$-valued functions on $X$. Brenner defined the \emph{continuous closure} $I^{\rm cont}$ of an ideal $I$ as $I\mathcal C(X) \cap R$. He also introduced an algebraic notion of \emph{axes closure} $I^{\rm ax}$ that always contains $I^{\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \in I^{\rm ax}$ if its image is in $IS$ for every homomorphism $R \to S$, where $S$ is a one-dimensional complete seminormal local ring. We also introduce the \emph{natural closure} $I^\natural$ of $I$. One of many characterizations is $I^\natural = I + \{f \in R: \exists n >0 \text{ with } f^n \in I^{n+1}\}$. We show that $I^\natural \subseteq I^{\rm ax}$, and that when continuous closure is defined, $I^\natural \subseteq I^{\rm cont }\subseteq I^{\rm ax}$. Under mild hypotheses on the ring, we show that $I^\natural= I^{\rm ax}$ when $I$ is primary to a maximal ideal, and that if $I$ has no embedded primes, then $I = I^\natural$ if and only if $I = I^{\rm ax}$, so that $I^{\rm cont}$ agrees as well. We deduce that in the polynomial ring $\mathbb C[x_1, \ldots, x_n]$, if $f = 0$ at all points where all of the ${\partial f \over \partial x_i}$ are 0, then $f \in ( {\partial f \over \partial x_1}, \, \ldots, \, {\partial f \over \partial x_n})R$. We characterize $I^{\rm cont}$ for monomial ideals in polynomial rings over $\mathbb C$, but we show that the inequalities $I^\natural \subset I^{\rm cont}$ and $I^{\rm cont} \subset I^{\rm ax}$ can be strict for monomial ideals even in dimension 3. Thus, $I^{\rm cont}$ and $I^{\rm ax}$ need not agree, although we prove they are equal in $\mathbb C[x_1, x_2]$.

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