Finding all homomorphisms

**AlexP** · May 9th 2011, 05:11 PM

My goal was to find all homomorphisms $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$ , and $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$ , and $\mathbb{Q} \to \mathbb{Q}$ . Here's what I have...

For $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$ I believe they are $(a,b) \mapsto 0$ , $(a,b) \mapsto (a,b)$ , $(a,b) \mapsto (b,a)$ , $(a,b) \mapsto (a,0)$ , and $(a,b) \mapsto (0,b)$ .

For $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$ I have the identity, the 0-map, $(a,b) \mapsto a$ , and $(a,b) \mapsto b$ .

And for $\mathbb{Q} \to \mathbb{Q}$ I just have the 0-map and the identity.

Did I get them all? And when there are more 'creative' ones like $(a,b) \mapsto (b,a)$ , how do we know when we've found them all?

**Deveno** · May 9th 2011, 06:44 PM

what about (a,b) ---> (ma,nb) and (a,b)--->(nb,ma)?

and for Z⊕Z -->Z, what about (a,b) --> ma+nb?

**AlexP** · May 9th 2011, 08:39 PM

Neither of the first two work unless n, m= 0 or 1 because upon finding $\phi\[(a,b)\]\phi\[(c,b)\]$ , there are $n^2$ and $m^2$ terms that do not appear with $\phi\[(a,b)(c,d)\]$ , which just has n and m.

The third one doesn't work since $\phi\[(a,b)(c,d)\]=mac+nbd$ and $\phi\[(a,b)\]\phi\[(c,b)\]=m^2ac+mand+nbmc+n^2bd$

The second does give $(a,b) \mapsto (b,0)$ and $(a,b) \mapsto (0,a)$ .

**Deveno** · May 9th 2011, 09:13 PM

perhaps you should indicate what kind of structure the homomorphisms are to preserve?

**Drexel28** · May 9th 2011, 09:20 PM

Agreed. Ring homomorphisms?

**AlexP** · May 10th 2011, 06:36 AM

Yes, ring homomorphisms. I suppose that would be good to know.

**Opalg** · May 10th 2011, 07:44 AM

Originally Posted by AlexP

My goal was to find all homomorphisms $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$ , and $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$ , and $\mathbb{Q} \to \mathbb{Q}$ . Here's what I have...

For $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$ I believe they are $(a,b) \mapsto 0$ , $(a,b) \mapsto (a,b)$ , $(a,b) \mapsto (b,a)$ , $(a,b) \mapsto (a,0)$ , and $(a,b) \mapsto (0,b)$ . (a,b) → (a,a) ? (a,b) → (b,b) ?

For $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$ I have the identity, the 0-map, $(a,b) \mapsto a$ , and $(a,b) \mapsto b$ .

And for $\mathbb{Q} \to \mathbb{Q}$ I just have the 0-map and the identity.

Did I get them all? And when there are more 'creative' ones like $(a,b) \mapsto (b,a)$ , how do we know when we've found them all?

A ring homomorphism has to take idempotents to idempotents. In the ring $\mathbb{Z} \oplus \mathbb{Z}$ , the only idempotents are (0,0), (1,0), (0,1) and (1,1). Also, every element of $\mathbb{Z} \oplus \mathbb{Z}$ is a linear combination of (1,0) and (0,1), so a homomorphism is specified by what it does to those two elements. That severely restricts the possible choices for a homomorphism.

**AlexP** · May 10th 2011, 01:05 PM

Thank you, that does clear things up. Ring homomorphisms are more rigid than I was thinking they are...I saw something like $(a,b) \mapsto (b,a)$ as being somewhat random and didn't know how it fit in with the properties of homomorphisms, but now I see what the basis for it is. Thanks to everyone else too.

Thread: Finding all homomorphisms

LinkBack

Thread Tools

Display

Finding all homomorphisms

Similar Math Help Forum Discussions

Homomorphisms

Finding all the homomorphisms

Homomorphisms to and onto

homomorphisms

Homomorphisms

Search Tags