Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the property that whenever an element $a \otimes b$ is zero under the natural map $$A_i \otimes_{A_0} A_j \to A_{i+j}$$ if and only if $i+j > N$. Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?