Exercise 10, Appendix 1

Suppose that $k$ is a field, that A is an infinite-dimensional $k$-algebra, and that every proper left ideal of A is finite-dimensional over $k$.

a) Prove that we have $A\mathfrak{a}= 0$ for every left ideal $\mathfrak{a}\not= A$ (consider the homomorphism $\gamma : A\rightarrow$ End$_k(\mathfrak{a})$).

b) Let $\mathfrak{r}$ be the sum of the maximal left ideals of A. Prove that $\mathfrak{r}$ is different from A (use Exercise 9), that $\mathfrak{r}$ is a two-sided ideal, and that $A/\mathfrak{r}$ is a field (cf. Exercise 7). c) Let $u$ be a (left or right) unit modulo $\mathfrak{r}$. Prove that we have $\mathfrak{r}u=\mathfrak{r}$ (observe that we have $Au= A$).

d) Prove that $\mathfrak{r}A$ is zero (consider the right annihilator of $\mathfrak{r}$ in A and reason as in a)) and then that $\mathfrak{r}$ is zero. Conclude that A is a field.