Abstract Algebra Answers

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.

Let A be an algebra over a field k such that every element of A is algebraic over k. Let B be a subalgebra of A, and b ∈ B. Show that b is a unit in B iff it is a unit in A.

Let A be an algebra over a field k such that every element of A is algebraic over k. Show that a nonzero element of A is a unit iff it is not a 0-divisor.

Let A be an algebraic algebra over a field k. Show that A is Dedekind-finite.

Show that any left noetherian ring R is Dedekind-finite.

Show that the left regular module R is hopfian iff R is Dedekind - finite.

Show that any noetherian module M is hopfian.

sqrt(x^2)=|x| how can i proof it by real analysis