## 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.
1. 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.
2. 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 [cx]m=[cy]m.
• 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.
3. 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.
4. Are Z/10Z and Z/2Z x Z/5Z isomorphic as rings? Why or why not?
5. Are Z/9Z and Z/3Z x Z/3Z isomorphic as rings? Why or why not?