Definition:Trace 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 $\operatorname{Tr}_{B/A}(b)$ of $b$ is the trace of the regular representation $\lambda_b : B \to B$ over $A$.

Also see

 * Trace Map of Algebra over Ring is Linear Form
 * Trace of Algebra over Algebra over Ring is Composition of Traces
 * Definition:Norm of Element of Algebra over Ring
 * Definition:Characteristic Polynomial of Element of Algebra