$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and Bordism Operations, available also on GitHub.

Let $R$ be a ring. We have a bijection $$\left\{\begin{gathered}\text{group scheme actions}\\\mathbb{G}_{m}\times\Spec(R)\to\Spec(R)\end{gathered}\right\}\cong\left\{\text{$\mathbb{Z}$-gradings of $R$}\right\}.$$ Moreover a ring map $f\colon A\to B$ respects the grading if $\Spec(f)\colon\Spec(B)\to\Spec(A)$ is $\mathbb{G}_{m}$-equivariant.

Question. Are there generalisations of this result for $A$-gradings with $A$ a commutative monoid? In particular, do we have such a result for $\mathbb{N}$-gradings?

$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and Bordism Operations, available also on GitHub.

Let $R$ be a ring. We have a bijection $$\left\{\begin{gathered}\text{group scheme actions}\\\mathbb{G}_{m}\times\Spec(R)\to\Spec(R)\end{gathered}\right\}\cong\left\{\text{$\mathbb{Z}$-gradings of $R$}\right\}.$$ Moreover a ring map $f\colon A\to B$ respects the grading if $\Spec(f)\colon\Spec(B)\to\Spec(A)$ is $\mathbb{G}_{m}$-equivariant.

Question. Are there generalisations of this result for $A$-gradings with $A$ a commutative monoid? In particular, do we have such a result for $(\mathbb{N},+,0)$-gradings? What about $(\mathbb{N},1,\cdot)$- or $(\mathbb{N}_{\geq1},1,\cdot)$-gradings?