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/bZ -> Z/mZ
by f([x]b) = [cx]m.
- Show that this function is well-defined; i.e., if
[x]b=[y]b, show that
- Show that this is a homomorphism of groups
if we regard Z/bZ and Z/mZ
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/10Z and Z/2Z x Z/5Z isomorphic as rings? Why or why not?
- Are Z/9Z and Z/3Z x Z/3Z isomorphic as rings? Why or why not?