Abstract Algebra Answers

the idealizer IR(A) and the eigenring ER(A) for the right ideal A = xR in the free k-ring R =
k<x, y>, and for the right ideal A = xR where x = i + j + k in the ring of quaternions with integer coefficients.

Let R = Mn(k) where k is a ring, and let A be the right ideal of R consisting of matrices whose first r rows are zero. Compute the idealizer IR(A) and the eigenring ER(A).

i have 51 skittle 15 yellow 6 red 13purple orange 17and green 12 i have to write the ratio of the number of skittles to mass of packet (grams)

Let R =
(Z Z2)
(0 Z2).
Show that R is not isomorphic to its opposite ring R^op

Let R =
(Z Z2)
(0 Z2).
Show that every right 0-divisor in R is a left 0-divisor.

For any ring k, let A = Mn(k) and let R (resp. S) denote the ring of n × n upper (resp. lower) triangular matrices over k. Show that R, S, R^op, S^op are all isomorphic.

For any ring k, let A = Mn(k) and let R (resp. S) denote the ring of n × n upper (resp. lower) triangular matrices over k. Suppose k has an anti-automorphism (resp. involution). Show that the same is true for A,R and S.

For any ring k, let A = Mn(k) and let R (resp. S) denote the ring of n × n upper (resp. lower) triangular matrices over k. Show that R is isomorphic to S.

Let R be a finite ring. Show that there exists an infinite sequence n1 < n2 < n3 < • • • of natural numbers such that, for any x ∈ R, we have x^n1 = x^n2 = x^n3 = • • • .

Let E = End(M) be the ring of endomorphisms of an R-module M, and let nM denote the direct sum of n copies of M. Show that End (nM) is isomorphic to Mn(E).