Proposition 8

Let $\Delta$ be a torsion-free commutative group and $A$ a graded ring of type $\Delta$. If the product in $A$ of two homogeneous elements $\neq 0$ is $\neq 0$, the ring $A$ has no divisor of 0.

Let $\Delta$ be given a total ordering compatible with its group structure (Lemma

1) and let $x = \sum_{\lambda \in \Delta} x_\lambda, y = \sum_{\lambda \in \Delta} y_\lambda$ be two non-zero elements of $A$ ($x_\lambda$ and $y_\lambda$ being homogeneous of degree $\lambda$ for all $\lambda \in \Delta$); let $\alpha$ (resp. $\beta$) be the greatest of the elements $\lambda \in \Delta$ such that $x_\lambda \neq 0$ (resp. $y_\lambda \neq 0$); it is immediate that if $\lambda \neq \alpha$ or $\mu \neq \beta$, either $x_\lambda y_\mu = 0$ or $\deg(x_\lambda y_\mu) < \alpha + \beta$; the homogeneous component of $xy$ of degree $\alpha + \beta$ is therefore $x_\alpha y_\beta$, which is non-zero by hypothesis; whence $xy \neq 0$.