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Math Help - Determine All Ring Homomorphisms: How to tackle such problems?

  1. #1
    Junior Member rubix's Avatar
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    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!
    Last edited by rubix; November 17th 2009 at 06:44 AM.
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  2. #2
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    Quote Originally Posted by rubix View Post
    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 f:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_{30} be a ring homom., then 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 \mathbb{Z}_{12}=\,<a> as an additive group (meaning: every element of the ring is of the form na for some n\in\mathbb{Z} and, in fact, we can choose 0\leq n\leq 11 since this

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

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

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

    Tonio
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