\begin{definition}[Definition:Norm of Element of Algebra over Ring]
Let $A$ be a commutative ring with unity.
Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module  over $A$.
Let $b \in B$.
The '''trace''' $N_{B/A}(b)$ of $b$ is the determinant of the regular representation $\lambda_b : B \to B$ over $A$.
\end{definition}