3 Let R S be isomorphic commutative rings with unity Prove t

3. Let R S be isomorphic commutative rings with unity. Prove the following: a). If R is an integral domain then S is an integral domain. b). If R is a field then S is a field 1s a 1 4. Find all the possible homomorphisms between the indicated rings

Solution

Solution : 3 )

a )

We wish to show that whenever s t = 0_S in S, then s = 0_S or t = 0_S. Let s, t S with s t = 0_S. Since f is onto, there exists a, b R such that f(a) = s and f(b) = t. So,

                                            s t = 0_S

                                   f(a) f(b) = 0_S

                                   f(a) f(b) = f(0_R)

                                        f(ab) = f(0_R).

Since f is one-to-one, ab = 0_R. Since R is an integral domain, a = 0_R or b = 0_R. Hence, s = f(a) = f(0_R) = 0_S or t = f(b) = f(0_R) = 0_S.

b )

Let s 0_S be in S. We wish to show s has a multiplicative inverse. Since f is onto, there exists a R such that f(a) = s. Since s 0_S we know a 0_R. Thus, since R is a field, a has a multiplicative inverse, a^-1 with

                                   a a^-1 = 1_R = a^-1 a

                              f(a a^-1) = f(1_R) = f(a^-1 a)

                            f(a) f(a^-1) = 1_S = f(a^-1) f(a)

                                s f(a^-1) = 1_S = f(a^-1) s.

Thus s is invertible with multiplicative inverse s^-1 = f(a^-1).