On the Nullstellens\"atze for Stein spaces and $C$-analytic sets

In this work we prove the real Nullstellensatz for the ring ${\mathcal O}(X)$ of analytic functions on a $C$-analytic set $X\subset{\mathbb R}^n$ in terms of the saturation of \L ojasiewicz's radical in ${\mathcal O}(X)$: The ideal ${\mathcal I}({\mathcal Z}({\mathfrak a}))$ of the zero-set ${\mathcal Z}({\mathfrak a})$ of an ideal ${\mathfrak a}$ of ${\mathcal O}(X)$ coincides with the saturation $\widetilde{\sqrt[\text{\L}]{{\mathfrak a}}}$ of \L ojasiewicz's radical $\sqrt[\text{\L}]{{\mathfrak a}}$. If ${\mathcal Z}({\mathfrak a})$ has `good properties' concerning Hilbert's 17th Problem, then ${\mathcal I}({\mathcal Z}({\mathfrak a}))=\widetilde{\sqrt[\mathsf{r}]{{\mathfrak a}}}$ where $\sqrt[\mathsf{r}]{{\mathfrak a}}$ stands for the real radical of ${\mathfrak a}$. The same holds if we replace $\sqrt[\mathsf{r}]{{\mathfrak a}}$ with the real-analytic radical $\sqrt[\mathsf{ra}]{{\mathfrak a}}$ of ${\mathfrak a}$, which is a natural generalisation of the real radical ideal in the $C$-analytic setting. We revisit the classical results concerning (Hilbert's) Nullstellensatz in the framework of (complex) Stein spaces. Let ${\mathfrak a}$ be a saturated ideal of ${\mathcal O}({\mathbb R}^n)$ and $Y_{{\mathbb R}^n}$ the germ of the support of the coherent sheaf that extends ${\mathfrak a}{\mathcal O}_{{\mathbb R}^n}$ to a suitable complex open neighbourhood of ${\mathbb R}^n$. We study the relationship between a normal primary decomposition of ${\mathfrak a}$ and the decomposition of $Y_{{\mathbb R}^n}$ as the union of its irreducible components. If ${\mathfrak a}:={\mathfrak p}$ is prime, then ${\mathcal I}({\mathcal Z}({\mathfrak p}))={\mathfrak p}$ if and only if the (complex) dimension of $Y_{{\mathbb R}^n}$ coincides with the (real) dimension of ${\mathcal Z}({\mathfrak p})$.