Noetherian $R$-algebra corresponds to a coherent sheaf of rings on $\operatorname{Spec}(R)$ Let $R$ be a ring and $A$ a Noetherian $R$-Algebra. Let $\newcommand{\m}{\mathcal} \m{A} = \tilde{A}$ be the corresponding $\m{O}_X$-Module, where $(X, \m{O}_X) = \operatorname{Spec}(R)$. I would like to show that $\m{A}$ then is a coherent $\m{A}$-Module on X.
Is the following attempt at a proof correct?
We first need that $\m{A}$ is of finite type, which is trivial. The nontrivial part is to prove that for every short exact sequence of $\m{A}$-Modules
$$
0 \to \m{K} \to \m{A}^n \to \m{A} \to 0
$$
the kernel $\m{K}$ is an $\m{A}$-Module of finite type.
Since $\m{A}$ and $\m{A}^n$ are quasi-coherent $\m{O}_X$-Modules, $\m{K} = \tilde{K}$ is quasi-coherent, with $K = \Gamma(X, \m{K})$. $K$ is an $\Gamma(X, \m{A})$-Module and by the equivalence of categories between $\mathsf{qCoh}(X)$ and $R$-$\mathsf{Mod}$, 
$$
0 \to K \to A^n \to A \to 0
$$
is an exact sequence (in $R$-$\mathsf{Mod}$, and hence also in $A$-$\mathsf{Mod}$). Since $A$ is Noetherian, it follows that $K$ is a $A$-Module of finite type, i.e. there is a surjection $A^m \to K \to 0$. Applying $\tilde{}$ gives an exact sequence
$$
\m{A}^m \to \m{K} \to 0
$$
(again, at first this is only exact in $\m{O}_X$-$\mathsf{Mod}$). $\square$
Background: Proposition 16.1.8 in EGA IV (part 4), which (I think) uses this fact.
(Grothendieck writes that this follows from the exactness of inverse images and $X = Y \times_{Y \times_S Y} (X \times_S Y)$, however I don't see how this is helpful at all. Alternatively - my french is not very good - these facts could be explaining why $\operatorname{gr}_I^\bullet A$ is Noetherian for $A$ Noetherian. If you can make sense of any of this please drop me a comment.)
 A: If $R$ is a commutative ring and $A$ a commutative $R$-algebra, it seems you may do the following: Let $X:=Spec(R)$. You want to define a sheaf of commutative $\mathcal{O}_X$-algebras on $X$. Let $\mathcal{O}:=\mathcal{O}_X$. For any open set $U \subseteq X$ you define $\mathcal{O}(U)$ as the limit over $R_f$ as $f\in A$ and $D(f)\subseteq U$. You may define $\mathcal{A}(U)$ as follows. View $A$ as a left $R$-module and define $\mathcal{A}(U)$ as the limit of $R_f\otimes_R A$ for $f \in R$ with $D(f)\subseteq U$. It follows $\mathcal{A}$ is a sheaf of $\mathcal{O}$-modules. Since tensor product commute with limits, it follows there is a canonical isomorphism $\mathcal{A}(U)\cong \mathcal{O}(U)\otimes_R A$. The $\mathcal{O}(U)$-module $\mathcal{A}(U)$ is canonically a commutative ring since it is the tensor product of commutative rings and all restriction maps are maps of rings, and it follows you have defined a sheaf of commutative $\mathcal{O}$-algebras $\mathcal{A}$ with the property that $\mathcal{A}(X)=A$.
Conversely given any quasi coherent sheaf of $\mathcal{O}$-algebras $\mathcal{A}$ it follows $\mathcal{A}(X):=A$ is a commutative $R$-algebra, whose sheafification gives back $\mathcal{A}$.
It seems: There is an equivalence of categories between the category of commutative $R$-algebras and maps of $R$-algebras and the category of sheaves of commutative $\mathcal{O}$-algebras and morphism of sheaves of commutative $\mathcal{O}$-algebras.
In the non-commutative case there are problems due to the fact that localization does not behave well for non-commutative rings. There is a class of rings where we can localize: rings of differential operators.
Example 0. The module of derivations. Let $\phi: A \rightarrow B$ be a map of commutative rings and let $T\subseteq A, S\subseteq B$ be  multiplicative subsets with $\phi(T) \subseteq S$. Let $\Omega^1_{B/A}$ be the module of Kahler differentials of $B/A$. There is a canonical isomorphism
$S^{-1}\Omega^1_{B/A}\cong \Omega^1_{S^{-1}B/T^{-1}A}$.
Hence when we dualize we get an isomorphism
$S^{-1}Der_A(B) \cong Der_{T^{-1}A}(S^{-1}B)$.
Hence the module of derivations localize to give a sheaf of $\mathcal{O}_Y$-modules on $Y:=Spec(B)$ relative to the canonical map $\pi: Y \rightarrow S:=Spec(A)$. Note that the localized module $S^{-1}Der_A(B)$ is canonically a $T^{-1}A$-Lie algebra. This localize and it follows the tangent sheaf $\Theta_{Y/S}$ (which is the sheafification of the module of derivations) is a sheaf of $\mathcal{O}_S$-Lie algebras. More precisely: it is a sheaf of $\mathcal{O}_S$-Lie-Rinehart algebras/Lie algebroids. This is a much studied subject.
Example 1: Jet bundles/principal parts. More generally if $\mathcal{P}^k_{B/A}:=B\otimes_A B/I^{k+1}$ is the $k$'th module of principal parts/$k$'th jet bundle of $B/A$, it follows
$S^{-1}\mathcal{P}^k_{B/A} \cong \mathcal{P}^k_{S^{-1}B/T^{-1}A}$.
When dualizing we get an isomorphism
$S^{-1}Diff^k_A(B) \cong Diff^k_{T^{-1}A}(S^{-1}B)$
where $Diff^k_A(B)$ is the module of $k$'th differential operators of $B/A$.
This gives a lozalization of the ring of differential operators $\mathcal{D}_A(B)$: There is an isomorphism
$S^{-1}\mathcal{D}_A(B) \cong \mathcal{D}_{T^{-1}A}(S^{-1}B)$.
Hence if we define for an open set $U \subseteq Y$ (with $\pi(U) \subseteq V$ where $V\subseteq S$ is an open set) $\mathcal{D}_{Y/S}^k(U)$ as the limit of $B_f \otimes_B Diff^k_A(B)$ for $D(f) \subseteq U$, it follows
$\mathcal{D}_{Y/S}^k(U)\cong Diff^k_{\mathcal{O}_S(V)}(\mathcal{O}_Y(U))$
Similarly we get
$\mathcal{D}_{Y/S}(U)\cong Diff_{\mathcal{O}_S(V)}(\mathcal{O}_Y(U))$.
It follows $(\mathcal{D}, \mathcal{D}^k)$ becomes a sheaf of filtered associative rings with the property that the associated graded ring is commutative. The structure sheaf $\mathcal{O}_Y$ is not in the center of $\mathcal{D}$.
Example 2: If $A=k$ is a field and $B:=k[x]$ is the polynomial ring in one variable it follows $\mathcal{D}_{B/A}$ is the Weyl algebra $W(1):=k[x,\partial]$ in one variable. Here $\partial$ is partial derivative wrto the $x$-variable. It follows $B$ is not in the center of $W(1)$ in general.
Example 3. Generalized ring of differential operstors. In general if $(U, U_i)$ is an $\mathbb{N}$-filtered associative ring  with $U_0:=B$ and $A$ in the center $c(U)$ such that the associated graded ring is commutative, we may localize. The filtered associative ring $(U,U_i)$ gives rise to a filtered sheaf $(\mathcal{U}, \mathcal{U}_i)$ of associative rings on $Y$ with $\mathcal{O}_S$ in the center, whose associative graded sheaf of rings is a sheaf of commutative $\mathcal{O}_Y$-algebras. Hence this type of "generalized ring of differential operators" localize well and give rise to sheaves of associative rings on $Y$. Many of the non-commutative rings appearing in mathematics/mathematical physics are of this type.
Example 4. We may view "non-commutative algebraic geometry" as the study of the following pairs: A non-commutative space is a pair $((Y,\mathcal{O}_Y), (\mathcal{U}, \mathcal{U}_i))$ where $(Y,\mathcal{O}_Y)$ is a scheme and $(\mathcal{U}, \mathcal{U}_i)$ is a sheaf of generalized differential operators in the above sense. There is a subject called "D-Lie algebras" where this notion is systematized. Such D-Lie algebras have many properties similar to Lie-Rinehart algebras/Lie algebroids/D-modules. We may introduce notions such as enveloping algebras, connections, Ext and Tor groups, characteristic varieties etc, and many of the results valid for Lie algebroids may be generalized to this new situation.
Example 5. Characteristic $p>0$. If $k$ is a field of characteristic $p$ and if $A:=k[X]$, it follows the ring $R:=k[x^p]\subseteq A$ is in the center of the ring $\mathcal{D}_{A/k}$. Hence in characteristic $p>0$ it follows the ring of differential operators has a non-trivial center.
