A graded ring is a ring $S$ together with a family $(S_{d})_{d\geq 0}$ of subgroups of the additive group of $S$, such that $S=\bigoplus_{d\geq 0}S_{d}$ and $S_{e}S_{d}\subseteq S_{e+d}$ for all $e,d\geq 0$.
I am having (and seems like always having) trouble to understand the part $S=\bigoplus_{d\geq 0}S_{d}$ as abelian groups.
Does this mean the following?
Every $s\in S$ can be written as $s=(x_{0},x_{1},x_{2},\cdots)$ where $x_{i}=x_{i}(s)$ is uniquely determinted by $s$, $x_{d}\in S_{d}$ for each $d$, and only finitely number of $x_{d}$ are nonzero.
If this is correct, why would many resources I find consider $s=\sum_{d\geq 0}x_{d}$, where $x_{d}=x_{d}(s)$ is uniquely determined by $s$, $x_{d}\in S_{d}$ and only finitely finitely number of $x_{d}$ are nonzero?
Are these two things the (essentially) same? If so, then I have trouble making them compatible with the ring multiplication.
For example, if we let $s,r\in S$, then we can write $s=(x_{0},x_{1},x_{2},\cdots)$ and $r=(y_{0},y_{1},y_{2},\cdots)$, so that $$sr=(x_{0}y_{0},x_{1}y_{1},x_{2}y_{2},\cdots),$$ so if the above two notions are equivalently, $sr$ should correspond to a sum $\sum_{d\geq 0}x_{d}y_{d}.$ However, if we write $s=\sum_{d\geq 0}x_{d}$ and $r=\sum_{d\geq 0}y_{d}$, then $$sr\neq \sum_{d\geq 0}x_{d}y_{d}.$$ Right?
This confuses me a lot. For example, in this post, Definition of graded rings, the argument is writing $1=\sum_{d\geq 0}x_{d}$ and then consider $x_{0}=x_{0}1=\sum_{d\geq 0}x_{d},$ which will give $x_{0}=x_{0}x_{d}$ for all $d\geq 0$.
However, if we do this analogously to vector notions, then we write $1=(x_{0},x_{1},\cdots, )$ so $$(x_{0},0,0,\cdots)=x_{0}=1x_{0}=(x_{0}x_{0}, x_{1}x_{0},\cdots),$$ which only gives us $$x_{0}=x_{0}x_{0}, x_{0}x_{d}=0\ \text{for all}\ d>0.$$
So in my view I do not think these notions coincide. Am I missing something here?
In "An introduction to the theory of groups", Rotman gives Lemma 10.4 in page 310:
Let $\{A_{k},k\in K\}$ be a family of subgroups of a group $G$, where $K$ is an index set. Then, the following are equivalent:
$G\cong\bigoplus_{k\in K}A_{k}$;
Every $g\in G$ has a unique expression of the form $g=\sum_{k\in K}a_{k}$ where $a_{k}\in A_{k}$, $k$ are distinct and $a_{k}\neq 0$ for only finitely many $k$.
$G=\langle \bigcup_{k\in K}A_{k}\rangle$ and for each $j\in K$, $A_{j}\cap\langle\bigcup_{k\neq j}A_{k}\rangle=0$.
So can I just consider $\bigoplus_{d\geq 0}S_{d}=\langle \bigcup_{d\geq 0}S_{d}\rangle$ such that $S_{d}\cap \langle \bigcup_{e\neq d}S_{d}\rangle=0$? Or For instance, the ideal $S_{+}:=\bigoplus_{d>0}S_{d}$ is just $\langle\bigcup_{d>0}S_{d}\rangle$ with the similar trivial intersection?
Yes, they are essentially the same. You may be seeing the distinction between an "external direct sum" and "internal direct sum", with you using the external sum notation, and the authors you are seeing utilizing the internal notation. Identify an element $x_d\in S_d$ with the image of $S_d$ in the (external) direct sum (i.e., the tuple whose $d$th entry is $x_d$, and every other entry is $0$).
The reason you are having trouble with the multiplication is that the direct sum is not a direct sum of rings, it is a direct sum of groups. I don't know if you've ever seen the construction of a polynomial ring that is done with formal direct sums, but it is the same idea here. You don't multiply the tuples entry-by-entry. Instead, you are supposed to have a multiplication defined on the entire direct sum which satisfies that $S_rS_k\subseteq S_{r+k}$.
For instance, let me expand on the construction of a polynomial ring alluded to above. What "is" the ring $\mathbb{Z}[x]$?
Well, I am going to define $\mathbb{Z}[x]$ as follows: its underlying set is the abelian group direct sum $$\bigoplus_{k=0}^{\infty} \mathbb{Z}=\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots.$$ Sum is done as usual: entry by entry.
Now I'm going to define a multiplication; it uses the multiplication I know from $\mathbb{Z}$, but it is not the multiplication entry by entry:
Given two elements $(a_0,a_1,a_2,\ldots)$ and $(b_0,b_1,b_2,\ldots)$ of the direct sum, I will define their product as $$(a_0,a_1,a_2,\ldots)\times(b_0,b_1,b_2,\ldots) = (c_0,c_1,c_2,\ldots)$$ where for each nonnegative integer $k$, $$c_k = a_0b_k + a_1b_{k-1}+\cdots+a_kb_0 = \sum_{j=0}^k a_jb_{k-j},$$ where the products and sum on the right hand side are the usual integer products.
We then define "$x$" to be the tuple $(0,1,0,0,\ldots)$, and for each $m\in\mathbb{Z}$, wee identify $m$ with the tuple $(m,0,0,\ldots)$. Then we prove that every element of $\mathbb{Z}[x]$ can be written uniquely as $$a_0+a_1x+\cdots+a_nx^n\quad \text{or as}\quad 0$$ where $a_n\neq0$, and the sums and products on the left hand side are the ones we've defined, not the ones from $\mathbb{Z}$.
Note this makes $\mathbb{Z}[x]$ into a ring (in fact a graded ring). But even though the elements are "really" tuples, we do not define the product entry-by-entry.
Rotman's definition is again trying to get at the fact that it is equivalent to describe an external direct sum as it is to describe an internal direct sum as subgroups of a group satisfying certain properties.