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Suppose that $R$ is a filtered algebra, that $r \in R_i$ and $s \in R_j$, and that these are respectively the smallest indexes which contain $r$ and $s$. Then by the axioms, $rs \in R_{i + j}$. Is there a name for when $i + j$ the minimal index $k$ for which $rs \in R_k$ (for all $r$ and $s$ in $R$)?

Now, if $R$ is just a regular ring considered to be filtered by $\{0\} \subseteq R \subseteq R \subseteq \cdots$, and $rs = 0$, then this doesn't hold. But if $R$ is an integral domain it does. And if $R$ is an integral domain, then the usual filtration on $R[x]$ satisfies this property as well. So it seems like this property might be related to being an integral domain. General references on this phenomenon would be well appreciated!

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    I don't know a name for this, but if I'm not mistaken it is equivalent to saying the associated graded ring is a domain. – 2017-02-23
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    Hi thank you that works for me! – 2017-02-24
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    OK, I'll turn that into an answer then. – 2017-02-24

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I don't know of any short name for this, but this condition is equivalent to saying the associated graded ring is a domain. Indeed, this condition says exactly that the product of two nonzero homogeneous elements of the associated graded ring is nonzero. This is equivalent to the product of any two nonzero elements of the associated graded ring being nonzero, since you can look at the highest-degree homogeneous parts of such elements.