Exercise 7, Appendix 1

We say that the $k$-algebra A is Artinian if every decreasing sequence of left ideals of A is stationary. The ring $\widetilde{A}$ is Artinian if and only if the algebra A and the ring $k$ are Artinian. Prove that if $k$ is a field, then an Artinian algebra A without radical is unital (observe that the algebra $\widetilde{A}$ is semisimple).