Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ and a graded homomorphism $\phi : R \to S$ , then there exists a graded homomorphism $\bar \phi : R[X]\to S$ such that $\bar \phi |_{R}=\phi$ and $\bar \phi (X)=a$ ? If this cannot always be done , then what if we took $a\in S$ to be a homogenous element ? If even this cannot always be done ; then is there any condition on the monoid $G$ such that this type of grading can be done for any $G$-graded ring ?
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1$\begingroup$ The way it's stated now, it's obviously false. If $a \neq 0$ is homogeneous of degree $g \in G$, then $X$ has to have a component of degree $g$. But this is only true for finitely many $g$. However, it should be possible if you fix the homogeneous degree of $X$ a priori (hence that of $a$ as well). $\endgroup$– R. van Dobben de BruynJun 24, 2017 at 14:54
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$\begingroup$ @R.vanDobbendeBruyn : you are right ; the homogenous degree of $X$ should be fixed a priori ; thanks $\endgroup$– user111524Jun 25, 2017 at 7:03
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This answer is only about the case where $G$ is a (commutative) group. I have not thought about more general monoids yet.
As observed by Remy in his comment, one has to choose $X$ to be homogeneous of some fixed degree and $a$ to be homogeneous as well. Then, this is indeed possible for polynomial algebras with an arbitrary set of indeterminates, or even more generally for algebras of monoids. For details you may have a look at paragraph 2.1.6 in this article (or here for free).
answered Jun 24, 2017 at 18:52
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$\begingroup$ could you please also take a look at this question math.stackexchange.com/questions/2331462/… ? $\endgroup$– user111524Jun 25, 2017 at 7:20
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$\begingroup$ Dear @users, the above also answers your question on MSE. The original version of that question (about a Graded Basis Theorem) is answered in this article (or here for free). $\endgroup$ Jun 25, 2017 at 10:39