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Module over a matrix ring

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data Let E be an n-dimensional vector space over a field k. Then if R is the ring of diagonal n-by-n matrices over k, E can be considered as a module over R, with the scalar multiplication diag(λ_1,...,λ_n)(a_1*e_1+...+a_n*e_n)=λ_1*a_1*e_1+...+λ_n*a_n*e_n, where e_1..._e_n form a basis for E as a vector space over k. Find the number of R-submodules of E.
    2. Relevant equations
    3. The attempt at a solution If W is a nontrivial R-submodule of E, then it is a k-vector subspace of E (this is trivial), so it has a basis w_1...w_m which can be extended to a basis w_1...w_n of E as a vector space over k. Then let the linear transformation T from E to E be defined by T(w_m)=w_m+1 and T(w_i)=0 for i not equal to m. Then T is diagonalizable, so it has an eigenbasis v_1...v_n for E, such that a subset, say v_1...v_m is a basis for W. With respect to the basis v_1...v_n, the matrix representation A of T is diagonal and thus A is an element of R. So we have Aw_m=w_m+1, which is not an element of W, so W is not closed under scalar multiplication with respect to R and thus W is not an R-submodule. Thus the only R-submodules of E are {1} and E.

    It turns out that my answer is wrong, and if you want I can provide a link to the correct solution. But where am I going wrong?
     
  2. jcsd
  3. Mar 17, 2012 #2
    Om, any thoughts on this?
     
  4. Mar 17, 2012 #3

    micromass

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    Why??

    Why??
     
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