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Non-orthogonal Beta variables

Let $k$ be a field and $A$ a $k$-algebra. Suppose that we have a left $A$-module $M$ and a $k$-algebra homomorphism $f:A\to \mathrm{End}(M)$. If $f(A)\subset \mathrm{End}_k(M)$, is it true that $f(A)$ is isomorphic to the matrix algebra $M_n(A)$ with $n$ being the rank of $M$?
The question if there is a general condition characterizing non-orthogonal subspaces of a vector space also arises. For a subspace $W$ of a $k$-vector space $X$ the following conditions are equivalent:

there is a $k$-vector space $Y$ such that $X=W\oplus Y$,
$W$ is non-orthogonal, that is, $\langle v,w\rangle=0$ for all $v\in W$ and $w\in W^\perp$,
$\dim_k W\le 1$.

A:

Yes. Let $R$ be the endomorphism ring of $M$. We

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