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The paper consider regulous functions on the real affine space$\mathbb{R}^N$. We shall study some algebraic properties of the ring of thosefunctions. It is presented a proof of the regulous version of Nullstellensatzbased on the substitution property and the Artin-Lang property for theconsidered function ring. We prove that every maximal ideal in the ring ofregulous functions on $\mathbb{R}^N$ when $N\geq 2$ is not finitely generated.Finally, we extend the latter result to an arbitrary, smooth, real affinealgebraic variety of dimension $d\geq 2$.

Maximal ideals in the ring of regulous functions are not finitely generated