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Suppose $V\neq0$ is a representation of an algebra A.

Definition: $v\in V$ is cyclic if and only if it generates $V$, thus $Av=V$. If a representation has a cyclic vector we call the representation cyclic.

I proved the following: $V$ irreducible iff all non-zero vectors of $V$ are cyclic (this follows immediatly from definitions).

But now i want to prove next result: $V$ is cyclic iff it is isomorphic to $A/I$ where $I$ is a left-ideal of $A$. But how to do this, i can not imagine an left ideal such that the representations $V$ and $A/I$ are isomorphic. We have thus to construct an intertwiner, but how?

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1 Answer 1

Hint Look at the homomorphism $A\to V, a\mapsto av$ for some cyclic vector $v$. Then use the isomorphism theorem.

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