I am trying to make sense of Definition 3.1 of the following paper. See my related question from earlier today. Here is the definition below:
Let $R$ be a graded ring and $I$ be a graded $R$-module. We say that $I$ satisfies the Baer condition if for each graded ideal $J$ in $R$, each $d\in\mathbb{Z}$ and each $R$-module homomorphism $\phi:\Sigma^d J\to I$, there exists $m\in I_d$ for which $\phi(a)=am$ for all $a\in J$.
(I assume $a\in I$ in the text is a typo).
I am a little puzzled by the role of $\phi$ in this. It is a map between two graded $R$-modules, and yet the authors do not insist that the morphism must respect the grading. Also $\Sigma^d J$ and $J$ are identical as $R$-modules, so the grading must come into this somewhere. What is going on here?