Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.


Then, given an exact sequence $\eta: 0\to X \to Z\_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $\eta$ by $f$ gives an exact sequence $f\eta: 0\to X \to Z'\_{\eta} \to M \to 0$ in $Ext^1(M,Y)$.


Similarly, given an exact sequence $\eta: 0\to M \to W\_{\zeta} \to Y \to 0$ in $Ext^1(Y,M)$, the pullback of $\zeta$ by $f$ gives an exact sequence $\zeta f: 0\to M \to W'\_{\zeta} \to X \to 0$ in $Ext^1(X,M)$.


It is well known that these assignments actually define functors $Ext^1(M,-); Ext^1(-,M): \text{Mod } R \to \text{Mod } R$, where for $f \in Hom(X,Y)$;


$$Ext^1(M,f): Ext^1(M,X) \xrightarrow{\eta \mapsto f\eta} Ext^1(M,Y)$$


$$Ext^1(f,M): Ext^1(Y,M) \xrightarrow{\zeta \mapsto \zeta f} Ext^1(X,M). $$


My question is: Are the functors $Ext^1(M,-)$ and $Ext^1(-,M)$ $R$-linear?


I think this should be well known, but I am looking for an explicit reference.


Please help.


Update: I have accepted Mark Grant's answer which gives a reference for additivity of the functor. If someone has an explicit reference for the fact that multiplication by ring elements is preserved too, please do post an answer; I'd really appreciate it and upvote your answer too!



Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
An explicit reference for the additivity is


*Mac Lane, Saunders*, Homology, Classics in Mathematics. Berlin: Springer-Verlag. x, 422 p. (1995). [ZBL0818.18001](https://zbmath.org/?q=an:0818.18001)


in particular Chapter III, Theorem 2.1. The book seems to be treating rings $R$ which need not be commutative, hence there is no mention of the $R$-module structure since it need not exist!


