So, there are two types of definitions of graded rings (I will consider only commutative rings) that I have seen:
1) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = \bigoplus_{n \in \mathbb{Z}} R_n$, where for all $m,n \in \mathbb{Z}, R_mR_n \subset R_{m+n}$.
2) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = \bigoplus_{n \in \mathbb{Z}} R_n$, where for all $m,n \in \mathbb{Z}, R_mR_n \subset R_{m+n}$, and $R_0$ is a subring of $R$, i.e., $1 \in R_0$.
In the second definition, is the additional condition that $R_0$ is a subring, i.e., basically the condition that $1 \in R_0$, redundant?