While reading about graded rings, I read that a graded ring $R$ is an infinite direct sum of abelian groups $\displaystyle R=\bigoplus_{i \in \mathbb Z} A_i$ together with a bilinear map $A_i\oplus A_j\longrightarrow A_{i+j}$.
I know that in the finite case we have that the direct sum of two abelian groups $A_1\oplus A_2$ is an abelian group and the addition is defined as $(a_1,a_2)+(a_1',a_2')=(a_1+a_1',a_2+a_2')$ but what if we take an infinite direct sum $\bigoplus_{i \in \mathbb Z} A_i$ of abelian groups, does the infinite case give an abelian group and what an element in this infinite direct sum looks like and how do we add two elements together?
and my last question is that is it always possible to construct a bilinear multiplication $A_i\oplus A_j\longrightarrow A_{i+j}$ especially when the $A_i $ are not subgroups of the same group but rather they are completely disctinct abelian groups put together into an infinite direct sum. Thank you for your clarification !!
It is simply the subgroup of all elements $(a_i)$ in the product $\displaystyle\prod_{i\in\mathbf Z}A_i$ such that all $a_i$ are $0$ but a finite number.
You may think of the construction of the ring of polynomials with coefficients in $R$; it is the $R^{(\mathbf N)}$ of sequences of coefficients almost all $0$.