Ext from a principal non-zero-divisor quotient (${\mathrm{Ext}}_A^0(A/(a),M)=\ker a,{\mathrm{Ext}}_A^1(A/(a),M)=M/aM$)

For a non-zero-divisor $a\in A$, the free resolution $0\to A\stackrel{a}{\to }A\to A/(a)\to 0$ gives degree-zero kernel and degree-one cokernel, and all higher Ext functors vanish. Directly in an injective resolution of the second variable, multiplication by $a$ is surjective in every term: the map $aA\to I$ taking $ax$ to $xm$ extends to $A\to I$. Thus $0\to {\mathrm{Hom}}_A(A/(a),I^{\bullet })\to I^{\bullet }\stackrel{a}{\to }{I}^{\bullet }\to 0$ is a short exact sequence of complexes and gives the same result. The displayed Hom functor is covariant in $M$; changing it to ${\mathrm{Hom}}_A(M,A/(a))$ changes the variable that is resolved.