What's an $\mathcal O_X$-algebra when $X= \operatorname{Spec} R$? Take $X= \operatorname{Spec}R$, $R$ a commutative ring with unit. What is an $\mathcal O_X$-algebra in that case? Is there more than just ordinary $R$-algebras?
Thank you in advance.

In the following thread
Noetherian $R$-algebra corresponds to a coherent sheaf of rings on $\operatorname{Spec}(R)$
you find the following: Let $X:=Spec(A)$ with $A$ a commutative unital ring. I believe the following "Theorem" holds:
"Theorem". There is an equivalence of categories between the category of sheaves of quasi coherent commutative $\mathcal{O}_X$-algebras and maps of $\mathcal{O}_X$-algebras and the category of commutative $A$-algebras and maps of $A$-algebras.
Proof. Given a commutative $A$-algebra $R$ and an open set $U \subseteq X$. Define
C1. $\mathcal{R}(U):=\mathcal{O}_X(U)\otimes_A R.$
It follows C1 defines $\mathcal{R}$ as a quasi coherent sheaf of commutative $\mathcal{O}_X$-algebras on $X$. Conversely given a quasi coherent sheaf of commutative $\mathcal{O}_X$-algebras $\mathcal{R}$, it follows $R:=\Gamma(X, \mathcal{R})$ is a commutative $A$-algebra. You may check that this defines an equivalence of categories.
In the non-commutative setting the situation becomes complicated due to the fact that you cannot localize non-commutative rings. There is a class of non-commutative rings that localize well - rings of differential operators.
