Let $M\subset {ℝ}^n$ be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider $S_K=f\in H\left(M\right)\left|f\right(x)\ne 0forallx\in K$; $S_K$ is a multiplicative subset of $H\left(M\right)$. Let $S_K^{-1}H\left(M\right)$ be the localization of H(M) with respect to $S_K$. In this paper we prove, first, that $S_K^{-1}H\left(M\right)$ is a regular ring (hence noetherian) and use this result in two situations:    1) For each open subset $\Omega \subset {ℝ}^n$, we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, $f_x$, is algebraic...