The condition ${\rm Ext}^1(M, R) = 0$ for modules over local Artin algebras $(R, \mathfrak{M})$ with $\mathfrak{M}^2 = 0$

Let $M$ be a finitely generated module over a (not necessarily commutative) local Artin algebra $(R,\mathfrak{M})$ with ${\mathfrak{M}^2} = 0$. It is known that when $R$ is Gorenstein (i.e. of finite injective dimension) $M = \sum R \oplus \sum R/\mathfrak{M}$. For $R$ not Gorenstein we describe all $M$ with ${\operatorname{Ext} ^1}(M,R) = 0$ and show that ${\operatorname{Ext} ^i}(M,R) = 0$ for some $i > 1$ if and only if $M$ is free. It follows that for $R$ not Gorenstein all reflexives are free. We also calculate the lengths of all the ${\operatorname{Ext} ^i}(M,R)$. As an application we show that if $(R,\mathfrak{M})$ is a commutative Cohen-Macaulay local ring of dimension $d$ which is not Gorenstein, if $R/{\mathfrak{M}^2}$ is Artin and $({x_1}, \cdots ,{x_d})$ is a system of parameters with ${\mathfrak{M}^2}$ contained in the ideal $({x_1}, \cdots ,{x_d})$ and if $M$ is a finitely generated $R$-module with ${\operatorname{Ext} ^i}(M,R) = 0$ for $1 \leqq i \leqq 2d + 2$, then $M$ is free.