Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a graded version for this? I mean:
Let $R$ be a graded ring. Let $M$ be a graded $R$-module.
If $Ext_R^1(R/I,E)=0$ for all homogeneous ideals $I$ of $R$ then $Ext_R^1(M,E)=0$ for every graded $R$-module $M$?
To get the correct graded version you have to observe all possible shifts. More precisely:
Let $G$ be an abelian group, let $R$ be a $G$-graded ring, and let $M$ be a $G$-graded $R$-module. Then, the following statements are equivalent:
(i) $M$ is injective;
(ii) For every $g\in G$, every monomorphism $v\colon N\rightarrow R(g)$ and every morphism $w\colon N\rightarrow M$ there is a morphism $u\colon R(g)\rightarrow M$ such that $u\circ v=w$.
(Note that in the above, "morphism" means morphism in the category of $G$-graded $R$-modules.)
In order to proof this you have to observe that the category of $G$-graded $R$-modules is an AB5-category and that $\bigoplus_{g\in G}R(g)$ is a generator of this category, and then you can apply Lemma 1 in Section 1.10 of Grothendieck's Tohoku paper (Link).
