$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's [Formal schemes and formal groups](https://arxiv.org/abs/math/0011121), and in Lemma 1.3.2 of Eric Peterson's [Formal Geometry and Bordism Operations](https://www.cambridge.org/br/academic/subjects/mathematics/geometry-and-topology/formal-geometry-and-bordism-operations?format=HB), available also on [GitHub](https://github.com/ecpeterson/FormalGeomNotes).

> 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?