Assume that $M = \bigoplus_{n = 0}^\infty M_n$ is a graded $A$-module, where $A = \bigoplus_{n = 0}^\infty A_n$ is a graded ring. We have by definition $A_m M_n \subset M_{m + n}$. Does this mean that for $a \in A_1$, $x \in M_1$ we have for instance $(0, a, 0, ...) \cdot (0, x, 0, ...) = (0, 0, ax, 0, ...)$? This seems quite odd to calculate.
Would this then mean that for $a \in A_2$, $x_i \in M_i$ we have $$(0, 1, a, 0, 0, ...) \cdot (0, x_1, x_2, 0, 0, ...) = (0, 0, x_1, x_2 \cdot a x_1, ax_2, 0, 0, ...)$$ (I assumed that two elements going to the same $M_i$ would again be multiplied.)
Thanks for any help!
You just apply the distributive property. The element $\alpha=(0,a,b,0,0,\dotsc)$ is just $a+b$ and $\xi=(0,x_1,x_2,0,0,\dotsc)$ is $x_1+x_2$; then $$ \alpha\xi=(a+b)(x_1+x_2)=ax_1+ax_2+bx_1+bx_2= (0,0,ax_1,ax_2+bx_1,bx_2,0,0,\dotsc) $$ because $ax_1$ has degree $2$, $bx_2$ has degree $4$ and both $ax_2$ and $bx_1$ have degree $3$.
It's no different at all than when you multiply polynomials and reduce alike terms.