Subrings and Ideals Questions

1) Which of the following subsets of M2(R) are subrings?

a) S = all matrices of the form

(0 b)

(c d)

(Sorry, I don't know how to do matrices. These are all each one matrix)

b) S = all matrices of the form

(a 0)

(c d)

c) S = GL(2, R)

d) S = all matrices of the form

(a b)

(b a)

2) Find a maximal ideal in

a) Z6

b) Z12

c) Z18

3) Let R = {q in Q s.t. q = a/b, a,b in Z and b is odd). Show that R has a unique maximal ideal.

4) Let R be a ring and I an ideal of R. Show that

a) if R is commutative, so is R/I

b) if R has a unity, so does R/I

5) Let R be a commutative ring.

a) Show that the set of nilpotent elements in R forms an ideal

b) Show that the quotient of R by this ideal has no nonzero nilpotent elements

6) Let R be an integral domain. We call R a **principal ideal domain (PID)** if every ideal of R is of the form aR for some a in R. Show that in a PID every nontrivial proper prime ideal is maximal.