Examples

1) Let $\mathbf{A}$ be a polynomial algebra $K[X_i]_{i \in I}$. For every extension $L$ of $K$ the ring $L \otimes_K \mathbf{A}$ is isomorphic to $L[X_i]_{i \in I}$ (III, p. 449, Remark 2) and hence is an integral domain (IV, p. 9, Prop. 8). In other words, every polynomial algebra over a field $K$ is a separable $K$-algebra.

2) Let $\mathbf{A}$ be a commutative algebra of finite degree over a field $K$. For $\mathbf{A}$ to be separable it is necessary and sufficient for it to be etale (V, p. 34, Th. 4).

3) Let $E$ be an algebraic extension of a field $K$. If $L$ is an extension of $K$, then the ring $L \otimes_K E$ is a union of the subrings $L \otimes_K F$ where $F$ ranges over the set of all subextensions of finite degree of $E$; therefore the ring $L \otimes_K E$ is reduced if and only if this is true of $L \otimes_K F$ for every subextension $F$ of $E$, of finite degree over $K$. Taking Example 2 into account we see that $E$ is a separable algebra in the sense of the above Def. 1 if and only if it is a separable algebraic extension in the sense of Def. 1 of V, p. 36.