Can you help me. Here is my work.

(a) Prove that there is no ring homomorphism Z/5Z → Z.

(b) Prove that there is no ring homomorphism Z/5Z→ Z/7Z.

(a) Axiom of ring homomorphism are

f(1) = 1

f(a+b) = f(a) + f(b)

f(ab) = f(a)*f(b)

Consequences:

f(0) = 0

f(-r) = -f(r)

Homomorphism must therefore be

f(0) = 0

f(1) = 1

f(4 mod 5) = f(-1 mod 5) = -f(1) = -1

f(4) = f(2) + f(2) = 2*f(2) = -1

f(2) = $\displaystyle -1 \cdot 2^{-1} $

f(2) can't exist because there is no inverse with respect to multiplication in Z.

(b)Homomorphism must therefore be

f(0) = 0

f(1) = 1

f(4 mod 5) = f(-1 mod 5) = -f(1) = -1 mod 7 = 6 mod 7

f(4) = f(2) + f(2) = 2*f(2) = 6

f(2) = $\displaystyle 6 \cdot 2^{-1} $ = 6 * 4 = 24 = 3 mod 7

f(2+3) = f(5) = f(0) = 0 = f(2) + f(3)

f(3) = -f(2) = -3 = 4 mod 7

Recapitulate

f(0) = 0

f(1) = 1

f(2) = 3

f(3) = 4

f(4) = 6

f(1+3) = f(4) = 6 not equal to f(1) + f(3) = 1+4 = 5

Contradiction and there is no homomorphism from Z/5Z→ Z/7Z