Do four of the book problems, two of the remaining problems. In this problem set, I use the typographical monstrosity [x] to denote tensor product. Sorry!

Book problems from section 10.3: 2, 3, 6, 12, 16, 17, 21.
Let M be the ring of 2x2 complex matrices. Let C be the left R-module of 2x1 complex matrices (the module structure is matrix multiplication). Let R be the right R-module of 1x2 complex matrices.
Compute, with proof, the tensor product of R and C over the ring M.
Comments: This gives an example of a tensor product over a non-commutative ring. Remember that a tensor product consists not only of an abelian group A, but a balanced map RxC ---> A. You need to specify both pieces of data.
Let K be a field and let V = K^n, W = K^m as K-modules. Show that:
- every element of V * W can be written as the sum of at most min(n,m) "pure tensors" v[x]w, for v in V, w in W;
- that there exist elements that cannot be written as the sum of fewer "pure tensors".
Comments: This exercise concerns the question asked in class: Is every element of a tensor product in the image of the corresponding balanced map? In this question, we referred to the image of this balanced map as "pure tensors." This is also called "simple tensor".
Let R be a commutative ring with 1.
Let f: R^2-->R^2, g: R^2-->R^2 be R-module homomorphisms; their tensor product gives a map f[x]g : R^2[x]R^2-->R^2[x]R^2. (See class notes or Theorem 13 of 10.3). We exhibited in class an isomorphism from R^2[x]R^2-->R^4; therefore, the tensor product gives a map h: R^4-->R^4.
Explain how to compute the 4x4 matrix of h, in terms of the 2x2 matrices of f and of g. For "fun" and not credit: how does the determinant of h relate to the determinants of f and g?
Comment: In this question we have used, as discussed in class, the fact that any homomorphism f: R^n-->R^m can be uniquely expressed by an nxm matrix. The rows of this matrix are given by f(1,0,...,0), f(0,1,0,...,0) and so on.