Proving that the set of units of a ring is a cyclic group of order 4
The set of units of is , how can I show that this group is cyclic?
My guess is that we need to show that the group can be generated by some element in the set, do I need to show that powers of some element can generate all elements in the other congruence classes?
For example , i.e. using 7 we can generate an element in the congruence class of 9, but can not generate 29 for example from any power of 7, so is it sufficent to say that an element is a generator if it generates at least one element in all other congruence classes?
Definition of polynomial ring
Given a ring R, the polynomial ring is defined as
However, it is not usually specified what x is. In order for multiplication to make sense, I guess it has to be an element in R at least. But is R[x] the set of all functions , or the set of those functions evaluated at x?
Sometimes R is required to be commutative. Does that make any difference for R[x]?