# Determine All Ring Homomorphisms: How to tackle such problems?

• Nov 17th 2009, 06:19 AM
rubix
Determine All Ring Homomorphisms: How to tackle such problems?
Let me start by saying i have missed pretty much the entire lecture on Group Homomorphism and probably more than a half of Ring Homomorphism so go easy on me.

I'm trying to catch up from the book...trying to do as much exercise as i can but i'm stuck on the very basic idea. How to determine all Group and Ring Homomorphism for a given mapping.

I know the properties of Homomorphism, theorems/lemmas related to it...and atm i can do problems like "given mapping, prove or disprove it is ring homomorphism".

Anyway, how does one go aboot approaching such problems? Am i supposed to use 1-> 1 property, if so how?

Here are some examples:

{ i would really appreciate if you gave a through explanation (along with your approach if possible) rather than just the solution }

1a) Determine all homomorphisms from Z_12 to Z_30

1b) Determine all ring homomorphisms from Z_12 to Z_30

2) Determine all ring homomorphisms from Z/6 to Z/2

3) Determine all ring homomorphisms from Q -> Q

4) Determine all ring homomorphism from R to R

Much appreciated!
• Nov 17th 2009, 06:47 AM
tonio
Quote:

Originally Posted by rubix
Let me start by saying i have missed pretty much the entire lecture on Group Homomorphism and probably more than a half of Ring Homomorphism so go easy on me.

I'm trying to catch up from the book...trying to do as much exercise as i can but i'm stuck on the very basic idea. How to determine all Group and Ring Homomorphism for a given mapping.

I know the properties of Homomorphism, theorems/lemmas related to it...and atm i can do problems like "given mapping, prove or disprove it is ring homomorphism".

Anyway, how does one go aboot approaching such problems? Am i supposed to use 1-> 1 property, if so how?

Here are some examples:

{ i would really appreciate if you gave a through explanation (along with your approach if possible) rather than just the solution }

1a) Determine all homomorphisms from Z_12 to Z_30

1b) Determine all ring homomorphisms from Z_12 to Z_30

2) Determine all ring homomorphisms from Z/6 to Z/2

3) Determine all ring homomorphisms from Q -> Q

Much appreciated!

I won't do these problems "thoroughly" but only remind you some very basic facts since, as you say, you already know the elementary stuff around this

subject: take problem (1) for instance, and let $\displaystyle f:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_{30}$ be a ring homom., then $\displaystyle f\left(\mathbb{Z}_{12}\right)<\mathbb{Z}_{30}\Long rightarrow f\left(\mathbb{Z}_{12}\right)$ must divide 30 AND ALSO 12(for example, because

any such ring homom. is also a group homom. and thus by Lagrange's theorem).

Now, since $\displaystyle \mathbb{Z}_{12}=\,<a>$ as an additive group (meaning: every element of the ring is of the form $\displaystyle na$ for some $\displaystyle n\in\mathbb{Z}$ and, in fact, we can choose $\displaystyle 0\leq n\leq 11$ since this

is just the ring of residues modulo 12), then $\displaystyle f$ is uniquely and completely determined by its value on $\displaystyle a$, so you've to count all the possible choices for the

image $\displaystyle f(a)\,\,o\!f\,\,a\,\,in\,\,\mathbb{Z}_{30}$ and s.t. the subgroup $\displaystyle <f(a)>\,\in\,\mathbb{Z}_{30}$ has order dividing 30 and 12...

For example, if $\displaystyle \mathbb{Z}_{30}=\,<b>$ as an additive group, then it CAN NOT BE that $\displaystyle f(a)=b\,\,\,or\,\,\,f(a)=7b$, since both $\displaystyle b\,,\,7b$ generate the whole $\displaystyle \mathbb{Z}_{30}$, but $\displaystyle f(a)=5b$ is fine since $\displaystyle \left|<5b>\right|=6$....etc.

Tonio