Doubt in the definition of graded rings

Question

My doubt is quite simple, I didn't understand what is the operation $\cdot$ in this definition:

Thanks

Answer 1

Given a ring $R$ and any two subsets $X\subseteq R$ and $Y\subseteq R$, we can define the subset $$X\cdot Y=\{x\cdot y\mid x\in X,y\in Y\}$$ where $x\cdot y$ is just the ordinary product of two elements of a ring. So the statement that $$S_d\cdot S_e\subseteq S_{d+e}$$ just says that the product of any element of $S_d$ with any element of $S_e$ is an element of $S_{d+e}$.