Abstract Algebra Answers

Ex. 4.1. In R, define a ◦ b = a + b − ab. Show that this binary operation is associative, and that (R, ◦) is a monoid with zero as the identity element.

Let k be a field of characteristic zero, and let R be the Weyl algebra A1(k) with generators x, y and relation xy − yx = 1. Let p(y) ∈ k[y] be a fixed polynomial. Show that R → End(Vk) is injective but not an isomorphism.

Let k be a field of characteristic zero, and let R be the Weyl algebra A1(k) with generators x, y and relation xy − yx = 1. Let p(y) ∈ k[y] be a fixed polynomial. Show that R • (x − p(y)) is a maximal left ideal in R, and that the simple R-module V = R/R • (x − p(y)) has R-endomorphism ring equal to k.

Let k be a field of characteristic zero, and (aij) be an m × m skew symmetric matrix over k. Let R be the k-algebra generated by x1, . . . , xm with the relations xixj − xjxi = aij for all i, j. Show that R is a simple ring iff det(aij) <> 0. In particular, R is always nonsimple if m is odd.

Let D be a division ring, V =(infinite direct sum)eiD, and E = End(VD). Show that the ring E has exactly three ideals for the case of E = End(VD), where dimDV = α is an arbitrary infinite cardinal.

Let D be a division ring, V =(infinite direct sum)eiD, and E = End(VD). Show that the ring E has exactly three ideals: 0, E, and the ideal consisting of endomorphisms of finite rank.

Show that over certain rings, the “rank” of a free module may not be defined.

Let R be a simple, infinite-dimensional algebra over a field k. Show that any nonzero left R-module V is also infinite-dimensional over k.

For a subset S in a ring R. Let R be a semisimple ring, I be a left ideal and J be a right ideal in R. Show that annl (annr(I)) = I and annr (annl(J)) = J.

Are the following sequences arithmetic? If so, what is the common difference and the explicit formula?
7) 183, 1835, 18355, 183555

8) -1, 1, 4, 8

9) -35, -42, -49, -56

10) 9, 0, -9, -18

11) 9, 17, 25, 33

12) 2, 5, 10, 17