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Chapter 5: Commutative Rings: Solved problems

1. Let R be the ring with 8 elements consisting of all 3 × 3 matrices with entries in Z2 which have the following form:

You may assume that the standard laws for addition and multiplication of matrices are valid.

(a) Show that R is a commutative ring (you only need to check closure and commutativity of multiplication).

(b) Find all units of R, and all nilpotent elements of R.

(c) Find all idempotent elements of R.

2. Let R be the ring Z2 [x] / < x2 + 1 >. Show that although R has 4 elements, it is not isomorphic to either of the rings Z4 or Z2 Z2.

3. Find all ring homomorphisms from Z120 into Z42.

4. Are Z9 and Z3 Z3 isomorphic as rings?

5. In the group Z180× of units of the ring Z180, what is the largest possible order of an element?

6. For the element a = (0,2) of the ring R = Z12 Z8, find Ann (a) = { r in R | ra=0 }. Show that Ann (a) is an ideal of R.

7. Let R be the ring Z2 [x] / < x4 + 1 >, and let I be the set of all congruence classes in R of the form [ f(x) (x2 + 1)].

(a) Show that I is an ideal of R.

(b) Show that R/I Z2 [x] / < x2 + 1 >.

Hint: If you use the fundamental homomorphism theorem, you can do the first two parts together.

(c) Is I a prime ideal of R?

8. Find all maximal ideals, and all prime ideals, of Z36 = Z / 36Z.

9. Give an example to show that the set of all zero divisors of a ring need not be an ideal of the ring.

10. Let I be the subset of Z[x] consisting of all polynomials with even coefficients. Prove that I is a prime ideal; prove that I is not maximal.

11. Let R be any commutative ring with identity 1.

(a) Show that if e is an idempotent element of R, then 1-e is also idempotent.

(b) Show that if e is idempotent, then R Re R(1-e).

12. Let R be the ring Z2 [x] / < x3 + 1 >.

(a) Find all ideals of R.

(b) Find the units of R.

(c) Find the idempotent elements of R.

13. Let S be the ring Z2 [x] / < x3 + x >.

(a) Find all ideals of S.

(b) Find the units of R.

(c) Find the idempotent elements of R.

14. Show that the rings R and S in Problem 12 and Problem 13 are isomorphic as abelian groups, but not as rings.

15. Let Z[i] be the subring of the field of complex numbers given by
Z[i] = { m+ni in C | m,n in Z } .
(a) Define : Z[i] -> Z2 by (m+ni) = [m+n]2. Prove that is a ring homomorphism. Find ker () and show that it is a principal ideal of Z[i].

(b) For any prime number p, define : Z[i] -> Zp [x] / < x2+1 > by (m+ni) = [ m+nx ] . Prove that is an onto ring homomorphism.

16. Let I and J be ideals in the commutative ring R, and define the function : R -> R/I R/J by (r) = (r+I,r+J), for all r in R.

(a) Show that is a ring homomorphism, with ker () = I J.

(b) Show that if I + J = R, then is onto, and thus R/(I J) R/I R/J.

17. Considering Z[x] to be a subring of Q[x], show that these two integral domains have the same quotient field.

18. Let p be an odd prime number that is not congruent to 1 modulo 4. Prove that the ring Zp[x] / < x2 + 1 > is a field.

Hint: Show that a root of x2 = -1 leads to an element of order 4 in the multiplicative group Zp×.