How is $R_iR_j$ defined in the definition of graded rings?

Question

In the definition of graded rings there is the condition $R_iR_j\subset R_{i+j}$, it involves the product of two subgroups $R_iR_j$.

How is the set $R_iR_j$ defined?

Edit: Is anyone aware of a textbook in which this is defined?

Answer 1

Here is an answer before the post was modified, asking for a reference.

The product of two subsets is defined exactly like the Minkowski sum: $$AB=\{ab\mid a\in A,b\in B\},$$ hence the condition $R_iR_j\subset R_{i+j}$ means that for every homogeneous elements $x,y\in R,$ the product $xy$ is homogeneous and $$\deg(xy)=\deg x+\deg y.$$

Answer 2

$R = \oplus_{i \in \mathbb{N}} R_i$ (as abelian groups), so there are natural inclusions $i_i : R_i \rightarrow R$. The condition is that if $a \in range(i_i), b \in range(i_j)$, then $ab \in range(i_{i+j})$