Lemma 4

Let $\pi$ be a nontrivial irreducible representation of A in E. For every nonzero element $\xi$ of E, one has $\pi (A)\xi = E$.

The subspace $\pi (A)\xi$ of E is stable under $\pi (A)$. Suppose that it is zero. Then the nonzero subspace $K\xi$ of E would be stable under $\pi (A)$, and hence equal to E; but this would imply that $\pi$ is the zero representation. Thus one has $\pi (A)\xi = E$.

Let $\pi$ be a nontrivial irreducible representation of A in E. By this lemma, the annihilator R of $\xi$ in A is a regular left ideal (A, VIII, p. 425, No.$^o1$) of A, and the representation $\pi$ is equivalent to the representation defined by the A-pseudomodule $A/R$. Since $\pi$ is irreducible, the ideal R is a regular maximal left ideal.