Below we have a definition of a graded $k$-algebra where $k$ is a field. I have a few questions. First, looking it up, there seems to be some ambiguity as to what $R$ is a direct sum of the $R_n$ as. Groups? $k$-modules? Is there anything from context that could tell me? Secondly, why is $R_0$ a subalgebra. If the $R_n$ are submodules, then for $r\in R_n$, $kr\in R_n$, and therefore its not hard to see that if $R$ is an integral domain, $k\in R_0$, but I can't see why this should work if $R$ is not an integral domain. Thanks in advance.
The phrase "where the subspaces..." indicates that we want to think of this as a direct sum of $k$-vector subspaces. The general definition of graded ring uses a direct sum of abelian groups, see https://math.unl.edu/~tmarley1/905notes.pdf for more details.
One can show without difficulty (see Remark 1.1 of the aforementioned link) that $1\in R_0$. Furthermore, since $R_0R_0\subseteq R_{0+0}=R_0$ we have that $R_0$ is a subring of $R$. Finally, combining this with the fact that $R_0$ is a $k$-subspace of $R$ we obtain that $R_0$ is indeed a $k$-subalgebra of $R$.