Abstract Algebra Answers

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).

Is left artinian domain must be a division ring?

Let R be a domain. If R has a minimal left ideal, show that R is a division ring.

For any ring k, let A subspace of M(2,k) where a11+a21=a12+a22. Show that A is a subring of M2(k), and that it is isomorphic to the ring R of 2 × 2 lower triangular matrices over k.