(adsbygoogle = window.adsbygoogle || []).push({}); Urgent!-Hmomorphism @:Z_2-->Z_4 &Z_2-->Z_6

1. The problem statement, all variables and given/known data

THis is probbably very easy, just i am kinda bogged down:

(a) Show that the mapping [tex]\theta:Z_2-->Z_6[/tex] with [tex] \theta(\bar 0)=\bar 0, \theta(\bar 1)=\bar 3[/tex]

Is an injective ring homomorphism?

(b) Show that there is no injective ring homomorphism [tex]\theta:Z_2-->Z_4[/tex]

Proof:

(a) Well, i said since [tex]\bar 0 =/=\bar 3[/tex] in Z_6 then such theta is injective.

Now to establish homomorphism, i proceded

[tex]\theta(\bar 0 \bar 1)=\theta(\bar 0)=\bar 0[/tex]

[tex]\theta(\bar 0)\theta(\bar 1)=\bar 0[/tex] so they are equal

[tex]\theta(\bar 0+\bar 1)=\thta(\bar 1)=\bar 3[/tex]

[tex]\thetea(\bar 0)+\theta(\bar 1)=\bar 3[/tex]

So, i concluded that we have a homomorphism

Well, this didn't cause me any problems, as far as my understanding goes. However, i am having trouble on the second part:

(b) I started like this:

In order for [tex]\theta:Z_2-->Z_4[/tex] to be a homomorphism, if [tex]\theta(\bar a)=\bar b, \bar a \in Z_2, \bar b \in Z_4[/tex] then we should have the following [tex]o(\bar b)|o(\bar a)[/tex] ( i think there is a theorem that says this)

So, looking at the orders of the elements in both rings, we notice that the only such possibility is:

[tex]\theta(\bar 0)=\bar 0, \theta(\bar 1)=\bar 2[/tex]

Now, i am failing to show that this is not an injective homomorphism.

Here is what i am doing:

[tex]\theta(\bar 0 \bar 1)=\theta(\bar 0)=\bar 0[/tex]

[tex]\theta(\bar 0)\theta(\bar 1)=\bar 0[/tex]

[tex]\theta(\bar 0 +\bar 1)=\theta(\bar 1)=\bar 2[/tex]

[tex] \theta(\bar 0)+\theta(\bar 1)=\bar 2[/tex]

So, by this reasoning it looks to me that such a mapping is a homomorphism.

However, now everything that i am trying is also 'showing' that such a theta is also injective.

SO, how to show that theta is not injective/????????

Many thanks in return!

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# Homework Help: -Hmomorphism @:Z_2->Z_4 &Z_2->Z_6

**Physics Forums | Science Articles, Homework Help, Discussion**