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.

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- Nov 25th 2012, 07:39 PMneedhelp2ring homomorphism
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. - Nov 25th 2012, 08:33 PMjakncokeRe: ring homomorphism
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