Proposition 4

Let $p = \bigoplus_{i \geq 0} p_i$ be a graded ideal of $A$; for $p$ to be prime, it is necessary and sufficient that, if $x \in A, \ y \in A,$ satisfy $x \notin p$ and $y \notin p,$ then $xy \notin p$.

The condition is obviously necessary. Conversely, if it is fulfilled, then in the graded ring $A/p = \bigoplus_{i \geq 0} A_i/p_i$ the product of two homogeneous elements $\neq 0$ is $\neq 0$ and hence $A/p$ is an integral domain (Algebra, Chapter II, § 11, no. 4, Proposition 7).