Let $R$ be a commutative ring, and $I$ an ideal in $R$. Then, we have a graded ring $R_{\ast}:= \bigoplus_{n\geq 0} I^n$, where $I^0=R$.
So, let $A$ be a polynomial ring with a valuable $x$ over $\mathbb{C}$ and set $A_{\ast}:=A\oplus (x) \oplus (x^2) \oplus (x^3) \oplus \cdots$, where $(x)$ is the ideal generated by $x$.
Then, $A_{\ast}$ is a graded ring, but I confused because how do we check $(x^n)(x^m)\subset (x^{n+m})$ in $A_{\ast}$ ? (Of course, $(x^n)(x^m)\subset (x^{n+m})$ in $A$.)
Moreover, the element $x\in A_{\ast}$, we have two expressions:
$x+0+0+\cdots = x = 0+x+0+\cdots$
Thank you.