Let M=⨁i∈ZCei be an infinite dimensional C-vector space, and let End(M) denote the C-algebra of C-linear endomorphisms of M. Let A and B be two commuting elements in End(M) satisfying the following condition: there exist integers m≤n<0<p≤q satisfying gd(−m,p)=gcd(−n,q)=1, and such that for every j∈Z, one has
Ae_j=\sum_{i=j+m}^{j+n} a_{i,j}e_i, \text{with } a_{i,j} \in \mathbb{C}, a_{j+m,j}a_{j+n,j} \ne 0,
Be_j=\sum_{i=j+p}^{j+q} b_{i,j}e_i, \text{with } b_{i,j} \in \mathbb{C}, b_{j+p,j}b_{j+q,j} \ne 0.
Let R⊂End(M) be the C-subalgebra generated by A and B. Note that R is commutative and M can be regarded as an R-module.(a) Show that R is an integral domain and is isomorphic to R≅C[x,y]/f(x,y), where f(x,y) is a non-zero polynomial such that f(A,B)=0.(b) Let K be the fractional field of R. Show that M⊗RK is a 1-dimensional vector space over K. algebracollege contestsmoduleendomorphism