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Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the following property: if $x \in A_i$ and $y \in A_j$ and $i+j \leq N$, then $$xy = 0 \text{ implies } x = 0 \text{ or } y = 0.$$ 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?

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  • $\begingroup$ To be clear, are you asking that $ab\neq 0$ when $a,b\neq 0$ or that $ab\neq 0$ when $a\otimes b\neq 0$? It might be the case that $a\otimes b=0$ in $A_i\otimes_{A_0}A_j$ even if $a$ and $b$ are both nonzero. $\endgroup$
    – Eric Wofsey
    Commented Nov 30, 2016 at 23:19
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    $\begingroup$ The first sentence is indeed unclear (whenever... iff ... sounds weird) $\endgroup$
    – YCor
    Commented Nov 30, 2016 at 23:22
  • $\begingroup$ @EricWofsey It should be $ab \neq 0$ when $a,b \neq 0$. $\endgroup$
    – Exit path
    Commented Nov 30, 2016 at 23:23
  • $\begingroup$ @EricWofsey I edited to drop the tensor, does this make more sense? $\endgroup$
    – Exit path
    Commented Nov 30, 2016 at 23:25
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    $\begingroup$ I took the liberty of editing the opening for clarity and conciseness (since I know the person who posed the question). $\endgroup$
    – Marty
    Commented Dec 1, 2016 at 0:21

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