In an $\Bbb{N}$-graded domain $A$, units are homogeneous 
Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish to show that if $a, a^{-1}\in A$, then $a\in A_0$.

What I've gotten so far is that we must have $a=\sum_n\,a_n$ and $a^{-1}=\sum_n\,b_n$, $\,a_n, b_n\in A_n \,\forall\,n$, and therefore $1=\sum_n\,\delta_n = \sum_n \sum_{i+j=n} a_i b_j \implies \sum_{i+j=n} a_i b_j = \delta_n$, so $a_0, b_0$ are units, but I can't get any farther than this.

Since $1 \in A_0$, every $\delta_n$ with $n \ge 1$ must be zero.  The highest degree $\delta_i$ will be the product of the highest degree $a_i$ and the highest degree $b_i$, and since $A$ is a domain this is non-zero.  Therefore since the degrees add the highest $a_i$ and $b_i$ must both be in degree $0$, i.e. $a$ and $a^{-1}$ are homogeneous degree $0$.
