consider op: R* → R+, q(x) = x² R* is the group of non-zero real numbers with respect to multiplication. R+ is the group of positive real numbers with respect to multiplication. Show that is an onto (surjective) homomorphism.

22. Let be a ring with finite number of elements. Show that the characteristic of divides .

Exercises 12. Prove that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. (Hint: Consider the mapping defined by for all .)

Prove that if f is a permutation on A, then (f1)1=f.

Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]

Prove that the set of all complex numbers that have absolute value forms a group with respect to multiplication.

7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.

14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.

Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?

Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .