Abstract Algebra Answers

Let E = End(Vk) where V is a right vector space over the division ring k. Let R be a subring of E and A be a nonzero ideal in R. Show that R is dense in E iff A is dense in E.

Let M be a (not necessarily finitely generated) semisimple left module over a semiprimary ring R, and let k = EndR(M). Show that the natural ring homomorphism R → End(Mk) is surjective.

Let M be a (not necessarily finitely generated) semisimple left module over a semiprimary ring R, and let k = EndR(M). Show that Mk is a finitely generated k-module.

Let R be a simple ring with center k (which is a field). Let x1, . . . , xn ∈ R be linearly independent over k. Show that, for any y1, . . . , yn ∈ R, there exist a1, . . . , am and b1, . . . , bm in R such that yi = (sum over j) ajxibj for every i.

A random sample of 1000 workers from south India shows that their mean wages are `47/- per week with a standard deviation of `28/-. A random sample of 1500 workers from north India gives a mean wage of `49/- per week with a standard deviation of `40/-. Is any there any significant difference between their mean level of wages?