Étale algebra

A commutative algebra $A$ finite-dimensional over a field $K$ for which the bilinear form induced by the trace $$\langle x,y \rangle = \mathrm{tr}_{A/K} (x\cdot y)$$ is non-singular. Equivalently, an algebra which is isomorphic to a product of fields $A \sim K_1 \times \cdots \times K_r$ with each $K_i$ a field extension of $K$.
Since $\langle xy,z \rangle = \mathrm{tr}(xyz) = \langle x,yz \rangle$, an étale algebra is a Frobenius algebra over $K$.