Definition:Norm of Element of Algebra over Ring

Definition
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 $\map {N_{B / A} } b$ of $b$ is the determinant of the regular representation $\lambda_b : B \to B$ over $A$.

Also see

 * Norm of Product of Elements of Algebra over Ring
 * Norm of Algebra over Algebra over Ring is Composition of Norms
 * Definition:Trace of Element of Algebra over Ring
 * Definition:Characteristic Polynomial of Element of Algebra