Thread: 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!

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 be a ring homom., then 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 as an additive group (meaning: every element of the ring is of the form for some and, in fact, we can choose since this

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

image and s.t. the subgroup has order dividing 30 and 12...

For example, if as an additive group, then it CAN NOT BE that , since both generate the whole , but is fine since ....etc.

Tonio