Ring Homomorphism Questions

**Janu42** · April 23rd 2009, 11:40 AM

1) Consider the mapping p: Z3 --> Z6 given by p(x) = 2x, for x = 0,1,2. Is p a ring homomorphism? How about the mapping p(x) = remainder of 4x (mod 6)?

**ThePerfectHacker** · April 23rd 2009, 08:50 PM

Originally Posted by Janu42

1) Consider the mapping p: Z3 --> Z6 given by p(x) = 2x, for x = 0,1,2. Is p a ring homomorphism? How about the mapping p(x) = remainder of 4x (mod 6)?

You first need to consider if $[x]_3\mapsto [2x]_6$ is well-defined. Say that $[x]_3 = [y]_3$ then $x\equiv y(\bmod 3)$ , so $2x\equiv 2y(\bmod 6)$ and hence $[2x]_6 = [2y]_6$ which means the mapping is well-defined. Now what about $[x]_3\mapsto [4x]_3$ ? Try to argue that this is also well-defined. To show it is a ring homorphism (depending on how you define "ring homomorphism") you just need to show $f(x+y) = f(x)f(y)$ and $f(xy) = f(x)f(y)$ . Notice $f( [x]_3) + f([y]_3)= [2x]_6 + [2y]_6 = [2(x+y)]_6 = f([x+y]_3)$ . Also, $f([x]_3)f([y]_3) = [2x]_6\cdot [2y]_6 = [4xy]_6 \not = [2xy]_6 = f([xy]_3)$ . Thus, it is not a homorphism, you try doing the second case.

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