Is $\Bbb Z_3$ a subset of $\Bbb Z$?

Question

I was told that this is not true. Based on my understanding of a subset it would seem that this is true.

If every element of $\Bbb Z_3$ is in $\Bbb Z$ then it is a subset. $0,1,2$ are in $\Bbb Z$, so is it not a subset?

Answer 1

This is not really the right question to ask, since you could always take the underlying set of $\mathbb{Z}/(3)$ to be some representative subset of $\mathbb{Z}$ (like $\{0,1,2\}$) and transfer addition and multiplication over, and then the underlying set of $\mathbb{Z}/(3)$ would be a subset of the underlying set of $\mathbb{Z}$. (The standard definition, of course, is in terms of equivalence classes, but in practice we take such representatives when performing concrete calculations.)

What really matters for algebra is that $\mathbb{Z}/(3)$ is not a subring of $\mathbb{Z}$. That is, no matter the particulars of how you actually define $\mathbb{Z}/(3)$ using set theory, there is no injective function $f:\mathbb{Z}/(3)\to\mathbb{Z}$ that preserves the addition and multiplication operations.