I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.

Assume that $\mathcal{R_X}$ is a sheaf of (not necessarily commutative) rings on a topological space. We also do not assume that $\mathcal{R}_X$ is noetherian or coherent. Let $D^b(\mathcal{R}_X)$ be the bounded derived category of $\mathcal{R}_X-Mod$. My question is: Is the subcategory $D^b_{coh}(\mathcal{R}_X)$ of pseudo-coherent complexes in $D^b(\mathcal{R}_X)$ defined? And if yes, does the definition of pseudo-coherent complexes of modules over commutative rings apply to the context of modules over sheaves of noncommutative rings?

In other words, if the notion $D^b_{coh}(\mathcal{R}_X)$ exists in this setting, can we view it as the (triangulated) subcategory of $D^b(\mathcal{R}_X)$ consisting of complexes which are quasi-isomorphic to bounded above complexes of locally free $\mathcal{R}_X$-modules of finite rank.

I append Joseph Lipman's notes "NOTES ON DERIVED FUNCTORS AND GROTHENDIECK DUALITY" (https://www.math.purdue.edu/~jlipman/Duality.pdf) where in the second paragraph of Section 4.3, he defines $D_{coh}^b(O_X)$ for the commutative structure sheaf $\mathcal{O}_X$ of a scheme $X$. Basically, I try to determine if his definition is general enough so that it goes true when one replaces $\mathcal{O}_X$ with a sheaf of noncommutative rings $\mathcal{R}_X$.