We fix a field F. Let R be a ring containing F.

WRITTEN INTERVIEW QUESTIONS PhD candidates should provide an authentic personal statement reflecting on their own personal interest. In the event that any outside resources are used, those should be c
June 6, 2020
A project is worth $15 million today without an abandonment option. Suppose the value of the project is $20 million one year from today with high…
June 6, 2020

We fix a field F. Let R be a ring containing F. We say that R is an algebra over F if addition and multiplication in R restrict to addition and multiplication in F and the element 1 is in F if the element 1 is in R. Let R be and algebra over F.1. Suppose that _1 and _2 are distinct homomorphisms F[x] –> R such that _1(c) = _2(c) = c for every c in the field. Show that _1(x) does not equal _2(x).2. If r is any element of R, show that there is a homomorphism _: F[x] –> R such that _(x) = r and _1(c) = _2(c) = _1 for every c in F.3. Show that there is a bijection between the set R and the set of homomophisms _: F[X] –> R such that _1(c) = _2(c) = c for every c in F