Show that the map f : Z9 maps to Z12 defined as x maps to 4x is a well defined ring
homomorphism. What is the kernel of this map. Find the quotient ring and
give it's multiplication table.
Well definition: Just check if, whenever x = y $\displaystyle \in Z_9 $ then $\displaystyle \phi(x) = \phi(y) \in Z_{12} $ Pretty straightforward
Kernel: What $\displaystyle x \in Z_9 $ such 4x is divisible by 12, Ker = {0, 3, 6}
Quotient Ring: $\displaystyle R / Ker(\phi) = \{0 + \{0, 3, 6\}, 1 + \{0, 3, 6\}, 2 + \{0, 3, 6\} \} $ so 3 elements in the quotient ring.
Multiplication Table
--0 1 2
0 0 0 0
1 0 1 2
2 0 2 1