Abstract Algebra Answers

Show that over certain rings, the “rank” of a free module may not be defined.

Let R be a simple, infinite-dimensional algebra over a field k. Show that any nonzero left R-module V is also infinite-dimensional over k.

For a subset S in a ring R. Let R be a semisimple ring, I be a left ideal and J be a right ideal in R. Show that annl (annr(I)) = I and annr (annl(J)) = J.

Are the following sequences arithmetic? If so, what is the common difference and the explicit formula?
7) 183, 1835, 18355, 183555

8) -1, 1, 4, 8

9) -35, -42, -49, -56

10) 9, 0, -9, -18

11) 9, 17, 25, 33

12) 2, 5, 10, 17

suppose 23% of the individuals in a population have a certain medical condition what is the probability 105 to 135 inclusive will have this in a population of 500

find the period of this equation
y=5sin((x/9)+(pi/6))

Let R be an n2-dimensional algebra over a field k. Show that R ∼ Mn(k) (as k-algebras) iff R is simple and has an element r whose minimal polynomial over k has the form (x − a1) • • • (x − an) where a1, . . . , an ∈ k.

Let R be any semisimple ring. Every element a ∈ R can be written as a unit times an idempotent.

Let R be any semisimple ring. If a ∈ R is such that I = aR is an ideal in R, then I = Ra.

Let R be any semisimple ring. Show that R is Dedekind-finite, i.e. ab = 1 implies ba = 1 in R.