Abstract Algebra Answers

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.

Let us call a ring A a matrix ring if A ∼ Mm(R) for some integer m ≥ 2and some ring R. True or False: “A homomorphic image of a matrix ring is also a matrix ring”?

Let R, S be rings such that Mm(R) ∼= Mn(S). Does this imply that m = n and R ∼ S?

R be a simple ring that is finite-dimensional over its center k, show that R is isomorphic to a matrix
algebra over its center k iff R has a nonzero left ideal A with (dimkA)2 ≤ dimkR

Let R be a simple ring that is finite-dimensional over its center k. Let M be a finitely generated left R-module and let E = End(RM). Show that (dimkM)2 = (dimkR)(dimkE).