Ore condition for graded rings

Definition 8.4.13 ([Lurie (2017), Definition 7.2.3.1, Remark 7.2.3.7]). Let \(A_*\) be a graded associative ring and let \(S \subseteq A_*\) be a set of homogeneous elements which contains the unit and is closed under multiplication. We say that \(S\) satisfies the left Ore condition if the following hold:

(a)

For every pair of elements \(x \in A_*\) and \(s \in S\), there exist elements \(y \in A_*\) and \(t \in S\) such that \(tx = ys\).

(b)

If \(xs = 0\) for some \(x \in A_*\) and \(s \in S\), then there exists some \(t \in S\) such that \(tx=0\).

It suffices to check these conditions for homogeneous \(x\), in which case \(y\) in part (a) may also be chosen homogeneous. An ordinary ring is regarded as concentrated in degree zero.