Exercise 11, § 7

Let K be a commutative field and V a vector space over K admitting a basis $(e_n)_{n\geqslant 1}$. Let $u$ and $v$ be the endomorphisms of V defined by $u(e_1) = 0,u(e_p) =e_{p-1}$ for $p >1$, and $v(e_p) =e_{p+1}$ for $p\geqslant 1$, and let A be the K-subalgebra of End$_K(V)$ generated by $1,u$, and $v$. We have $uv= 1_V$ and $vu\not= 1_V$.

a) Prove that the elements $v^iu^j(i\geqslant 0, j\geqslant 0)$ form a basis of A over K.

b) Prove that the ring A is primitive and that its socle $\mathfrak{s}$ is the set of endomorphisms $w$ of V such that the sequence $(w(e_p))_{p\geqslant 1}$ has finite support (observe that $\mathfrak{s}$ is the smallest two-sided ideal of A containing $1_V-vu$). Prove that the K-algebra $A/\mathfrak{s}$ is isomorphic to $K[X,X^{-1}]$.