Abstract Algebra Answers

Show that, if R has an identity 1, the map ϕ : (R, ◦) → (R,×) sending a to 1 − a is a monoid isomorphism. In this case, an element a is left (right) quasi-regular iff 1 − a has a left (resp. right) inverse with respect to ring multiplication.

Show that any nilpotent element is quasi-regular in every ring.

Show that if ab is left quasi-regular element of ring , then so is ba.

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.

1. Consider the function f(x)=x^3-2
using Newton's method. Take x0=1.5 for the starting value.

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.