Remark 2

Suppose that A is an Artinian algebra over the field K. If A is commutative, then we can show (VIII, p. 180, Exercise 9) that A is isomorphic to a product of algebras $A_1\times \cdots \times A_n$ such that $A_i/\mathfrak{R}(A_i)$ is a field for every $i$. Conversely, if A is not commutative, then A may not be isomorphic to a product of algebras $A_1\times \cdots \times A_n$ such that $A_i/\mathfrak{R}(A_i)$ is a simple ring for every $i$ (VIII, p. 247, Exercise 5).