Definition 2

We call the homological dimension of $A$ and denote by $\operatorname{dh}(A)$ the supremum in $\mathbf{Z}$ of the set of integers $n$ for which there exist two $A$-modules M and N such that $\operatorname{Ext}_A^n(M,N)\neq 0$.

We have $\operatorname{dh}(0)=-\infty$, $\operatorname{dh}(A)\geq 0$ if $A\neq 0$. We shall see below that $\operatorname{dh}(A)=1$ if $A$ is principal and is not a field, and that, if K is a commutative field