Krdytkyu

Suppose $R$ is a (commutative) ring, and $M$ and $N$ are (finitely generated) $R$-modules. Then I know each $mathrm{Ext}_R^i(M, N)$ has the structure of an $R$-module. On the other hand, via Yoneda's description of Ext, each $varepsilon in mathrm{Ext}_R^i(M, N)$ corresponds to an equivalence class of exact sequences starting with $N$ and ending with $M$. My question is this: suppose $r in R$ and $varepsilon in mathrm{Ext}_R^i(M, N)$. How can I understand $r varepsilon$ in terms of $varepsilon$? To what extension does $r varepsilon$ correspond?

we can just see the case $i=1$:
In general,if $M$ is $A-B$ bimodule(i.e.left $A$ module and right $B$ module and $(am)b=a(mb)$),N is $A-C$ bimodule,then $Ext^1(M,N)$ is a $B-C$ bimodule.the structure of left $B$ and right $C$ module as follows:

if $delta:0rightarrow Nxrightarrow f Xxrightarrow g Mrightarrow 0$ is a short exact sequence in left $A$-modules.$forall bin B$,there is a left $A$-module homomorphism $varphi_b:Mrightarrow M$ by senting $m$ to $mb$.take pullback with $varphi_b$ and $g$,we get an element in $Ext^1(N,M)$ this is $bcdot delta$.

similarly,if $forall cin C$,take pushout with natural right multiplication $psi_c:Nrightarrow N$ and $f$,we get an element in $Ext^1(N,M)$ this is $deltacdot c$.
it is easy to Check this: structre of left $B$ module and right $C$ module has associativity.
As follows:
Then $varphi_a=alpha^-$ is using the unique map induced by kernel.

Now we consider the commutative case,we only need to check $rcdot varepsilon=varepsiloncdot r$,suppose $varepsilon:0rightarrow Nxrightarrow f Xxrightarrow g Mrightarrow 0$ is
a short exact sequence in $R-Mod$.Since $R$ is commutative ring,we have pullback diagram:
so $varepsilon= rcdotvarepsilon$,similarly $varepsilon =varepsiloncdot r$.we are done.

for $i>1$,the same.