## Math 310: Problem set 7

*Instructions*: This problem set is due
Thursday, November 2, 2006.
Your goal is not only
to give correct answers but to communicate
your ideas well. Make sure you use good English.
- Let R be a commutative ring such that 1 and 0 are not equal.
If r is a unit in R, then show that r is not a 0-divisor.
- Let b and c be integers bigger than 1, and let m=bc.
Define the function f:
**Z**/b**Z** -> **Z**/m**Z**
by f([x]_{b}) = [cx]_{m}.
- Show that this function is well-defined; i.e., if
[x]
_{b}=[y]_{b}, show that
[cx]_{m}=[cy]_{m}.
- Show that this is a homomorphism of groups
if we regard
**Z**/b**Z** and **Z**/m**Z**
as groups under addition.
- Determine whether f is also a homomorphism of rings.
Justify why or why not.

- Let s denote the square root of 3, hence about 1.732...
Let
**Q**[s] denote the set of all real numbers of the form
h(s), where h is a polynomial with rational coefficients (i.e., h is an element of **Q**[x]). Note that h(s) simplifies to a number of the form
us+v, where u and v are rational.
- Is
**Q**[s] a ring? Justify.
- Is
**Q**[s] a field? Justify.

- Are
**Z**/10**Z** and **Z**/2**Z** x **Z**/5**Z** isomorphic as rings? Why or why not?
- Are
**Z**/9**Z** and **Z**/3**Z** x **Z**/3**Z** isomorphic as rings? Why or why not?