# ring homomorphism

• November 25th 2012, 07:39 PM
needhelp2
ring homomorphism
Show that the map f : Z9 maps to Z12 defi ned as x maps to 4x is a well de fined ring
homomorphism. What is the kernel of this map. Find the quotient ring and
give it's multiplication table.
• November 25th 2012, 08:33 PM
jakncoke
Re: ring homomorphism
Well definition: Just check if, whenever x = y $\in Z_9$ then $\phi(x) = \phi(y) \in Z_{12}$ Pretty straightforward
Kernel: What $x \in Z_9$ such 4x is divisible by 12, Ker = {0, 3, 6}
Quotient Ring: $R / Ker(\phi) = \{0 + \{0, 3, 6\}, 1 + \{0, 3, 6\}, 2 + \{0, 3, 6\} \}$ so 3 elements in the quotient ring.
Multiplication Table

--0 1 2

0 0 0 0

1 0 1 2

2 0 2 1