Cotangent space of a local ring

Remarks

The scalar multiplication is well defined because $\mathfrak m \cdot \mathfrak m \subseteq \mathfrak m^2$.

Example: let $k$ be a field and $R = k[x, y]_{(x, y)}$, the localization of the polynomial ring at $(x, y)$. Then $\mathfrak m = (x, y)R$ and $\mathfrak m / \mathfrak m^2$ has the basis given by the classes of $x$ and $y$, so it has dimension $2$. For the cusp $R = k[x, y]_{(x, y)} / (y^2 - x^3)$, the classes of $x$ and $y$ still form a basis, because $y^2 - x^3 \in \mathfrak m^2$; the dimension is $2$ although the cusp is a curve.

Non-example: the quotient $\mathfrak m / \mathfrak m^3$ is in general not a vector space over $k$. In $k[\![t]\!]$ with $\mathfrak m = (t)$, the element $t$ acts on $\mathfrak m / \mathfrak m^3$ by sending the class of $t$ to the nonzero class of $t^2$, so $\mathfrak m$ does not kill it.