# Math Help - Subrings and subgroups of Z

1. ## Subrings and subgroups of Z

In Dummit and Foote Ch. 10 Introduction to Rings on page 228 we read:

"2Z is a subring of Z, as is nZ for any integer n. The ring Z/nZ is not a sub-ring (or a subgroup) of Z for any n $\geq$ 2."

Can anyone help me explictly prove that the ring Z/nZ is not a sub-ring (or a subgroup) of Z for any n >= 2.

Peter

2. ## Re: Subrings and subgroups of Z

Wouldn't this be since the (nontrivial) quotient group isn't even a subset of Z?

3. ## Re: Subrings and subgroups of Z

Hmmm ... yes, its a thought, definitely

But then the example of D&F is a little peculiar!

I was thinking that they were identifying $[a]_n$ with a, but of course if you do this you appear to have a subring (i think)

I tried this with Z/3z and it appeared to be both a subgroup and a subring of Z - so maybe you are correct.

Peter

4. ## Re: Subrings and subgroups of Z

you need to show that there is no (injective or even non-zero) group homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}.$ to see this just look at the image of $[n]=[0].$

5. ## Re: Subrings and subgroups of Z

Thanks.

To clarify, are you essentially saying that because there are n elements in Z/nZ and an infinite number in Z there can be no injective mapping? Or is there more to it?

[I guess I am finding it hard to figure out why "you need to show that there is no (injective or even non-zero) group homomorphism from Z/nZ to Z" (post by NonCommAlg - thanks) leads to the conclusion that "the ring Z/nZ is not a subring (or a subgroup) of Z"]

I started looking at the idea of using a homomorphism to establish what we need and was bothered by the following:

In Dummit and Foote section 7.3 we find:

--------------------------------------------------------------------------------------------------------------------------------------
"Proposition 5: Let R and S be rings and let $\phi$ : R $\rightarrow$ S be a homomorphism
(1) The image of $\phi$ is a subring of S
(2) the kernel of $\phi$ is a subring of S"
--------------------------------------------------------------------------------------------------------------------------------------

Now, consider the rings Z and Z/2Z

The map $\phi$: Z $\rightarrow$ Z/2Z defined by $\phi$: (even integer) ---> $[0]_n$ and $\phi$: (odd integer) ---> $[1]_n$ is a homomorphism.

Thus its image (which is Z/2Z) is a subring

What have I got wrong? This does not square with D&Fs statement that "2Z is a subring of Z, as is nZ for any integer n. The ring Z/nZ is not a sub-ring (or a subgroup) of Z for any n $\geq$ 2."