Questions Involving Rings

1) Let R be a ring with unity 1. Show that (-1)a = -a for all a in R.

2) Show that the set of all real numbers of the form a+bsqrt(2), where a,b in Q, forms a field under ordinary addition and multiplication.

3)

a) Show that every element of Zn (integers mod n) is either a unit or a zero-divisor. (Use the fact that an element of this ring is a unit iff (a,n) = 1.)

b) Which elements of Zn are nilpotent?

4) Show that if R is a ring with unity, then the multiplicative identity element in R is unique.

5) Let R be a ring with unity. R is called **Boolean** if every element of R is idempotent. Show that if R is Boolean then:

a) 2r = 0 for every r in R (in other words, r = -r) (Hint: Consider (r+r)^2)

b) R is commutative.