- Let R be the ring
**Z**/(m) and consider the ring S = R[x]/(x^{2}-x+1).- (a) Show that every element of S is the image (under the quotient homomorphism) of an element of the form ax + b in R[x], for some a and b in R.
- (b) In each case (i.e., m=2 and m=3), determine the product (x+1)(x-1) in S (i.e., write the product as ax+b for some a and b), and find the inverse of x-1 if it exists.
- (c) Determine all ideals of S, if m = 2 (explain). Is S a field? Why or why not?
- (d) Determine all ideals of S, if m = 3 (explain). Is S a field? Why or why not?

- Let P be an ideal in a (nonzero commutative) ring R (with 1).
- (a) A proper ideal with either of the following three properties
is called a
*prime*ideal. Show that the properties are equivalent:- (i) If I and J are ideals in R and IJ is contained in P, then either I is contained in P or J is contained in P.
- (ii) If a and b are in R with ab in P, then either a is in P or b is in P.
- (iii) The quotient R/P is a domain.

- (b) Show that a maximal ideal is prime.

- (a) A proper ideal with either of the following three properties
is called a
- A subset S of a domain R, such that S is closed under
multiplication and contains 1 but does not contain 0 is said to be a
*multiplicative*set. Let F be the field of fractions for R.- (a) If P is a prime ideal, prove that S = R-P is a multiplicative set.
- (b) If r is a nonzero element of R, prove that
S = {1, r, r
^{2}, ...} is a multiplicative set. - (c) Let S be a multiplicative
set. The subset of F of all elements of F of the form
a/s where s is in S, forms a subring of F denoted
S
^{-1}R. (You may assume this, and that R is a subring of S^{-1}R. In the case that S= R-P for a prime ideal P, the ring S^{-1}R is often denoted R_{P}. In the case that S = {1, r, r^{2}, ...} for some nonzero element r of R, the ring S^{-1}R is often denoted R_{r}.)- (i) Find S
^{-1}R in case R =**Z**, F =**Q**, and S =**Z**- (2). - (ii) Find S
^{-1}R in case R =**Z**, F =**Q**, and S = {1, 2, 2^{2}, ...}. (Elements of S^{-1}R are called dyadic rationals.)

- (i) Find S

- Let R denote the polynomials
**C**[x] with complex coefficients. Let f be a nonzero polynomial in R.- (a) Let I be an ideal in R
_{f}, and let J be the intersection of I with the subring R. Show that J is an ideal, hence principal and so of the form J = gR for some g in R; show that I = gR_{f}. - (b) Let h be in R. Show that h is invertible in
R
_{f}if and only if every root of h is a root of f. If h = x-c for some complex number c, show that hR_{f}is a maximal ideal if and only if f(c) is not 0.

- (c) Show that there is a bijection between the points
c of the complex plane
**C**such that f(c) is not 0, and maximal ideals in R_{f}. (This explains the terminology "localizing" for adding invertible elements. If you localize by inverting a polynomial f with more roots, the maximal ideals of R_{f}correspond to a smaller open subset of the complex plane**C**.)

- (a) Let I be an ideal in R
- If R and S are rings then so is RxS, where
(a,b)*(c,d) = (ac,bd) and (a,b)+(c,d) = (a+c,b+d).
Find a ring isomorphism
__f__:**R**[x]/(x^{2}-1) ->**R**x**R**, where**R**denotes the reals. Prove that your choice of f is indeed an isomorphism. [Hint: To define__f__, first define an appropriate homomorphism f:**R**[x] ->**R**x**R**.]