Assume $k$ is a field and $A$ is a $k$-algebra then the Wedderburn decomposition says $A=A_{sep}\oplus Nil(A)$, $A_{sep}\rightarrow A_{red}$ via $a\mapsto \bar{a}:=a+Nil(A)$. Where $A_{sep}$ means separable part of $A$ over $k$ and $A_{red}$ means $\frac{A}{Nil(A)}$ and $Nil(A)$ is nilpotents of $A$.
I know that $A_{sep}$ contains elements of $A$ which are separable over $k$ meaning that their minimal polynomial over $k$ is separable. I looked for the Wedderburn decomposition but I encounter to things related to groups. Can anyone suggest me the simplest reference for this theorem and its proof?