On commutative unital graded ring in which no non-zero homogenous element has a zero divisor in the homogenous part

Question

A follow up of On commutative unital graded rings in which no element in any homogenous part has a zero divisor . Let $G$ be a torsion free abelian group , let $R$ be a commutative unital $G$-graded ring such that for every $g \in G$ , $x_gy_g \ne 0 , \forall x_g,y_g \in R_g \setminus \{0\}$ ; then is it true that $R$ is an integral domain ? If not true , then does any other extra condition on $G$ implies the statement is true for any such $G$-graded ring ?

Answer 1

No. For instance, let $G=\mathbb{Z}$, let $k$ be a domain, and let $R=k[x,y]/(xy)$ graded such that every element of $k$ has degree $0$, $x$ has degree $1$, and $y$ has degree $-1$. A homogeneous element of degree $n$ is then of the form $ax^n$ for $a\in k$ if $n\geq 0$, or $ay^{-n}$ for $a\in k$ if $n\leq 0$. The product of any two nonzero such elements in the same degree will then always be nonzero. But $R$ is not a domain since $xy=0$.

Since $\mathbb{Z}$ embeds in every nontrivial torsion-free abelian group, this shows that there is no additional condition you could impose only on $G$ (besides $G$ being trivial) on top of what you are assuming which would make the answer yes.