Thread: Describe all units and zero divisors in the set of integers

1. Describe all units and zero divisors in the set of integers

Hi, I posted a little while ago asking about questions from the book Numbers and Symmetry by Bernard Johnston (Circular Numbers). I am still having a little trouble with the questions concerning these "circular numbers":

1. What is $\displaystyle \mathbb{Z}_{\infty}$?
I am unclear on how to answer this because I am unclear on what exactly is meant by $\displaystyle \mathbb{Z}_n$. The book prefers a rather indirect approach to the topic. Previously I guessed that it is equal simply to $\displaystyle \mathbb{Z}$ (the regular integers), or maybe just the nonnegative integers.

2. Describe all units and zero divisors in $\displaystyle \mathbb{Z}$.
I find it difficult to answer this question seeing as I can't even answer the first one. If $\displaystyle \mathbb{Z}_{\infty}$ is indeed equal to the set of nonnegative integers, then I think all the units in it would be such members as multiply to $\displaystyle \infty - 1$, which I assume is just $\displaystyle \infty$ and therefore there are no such members. Likewise, zero divisors would be such members as multiply to zero, which is the same as $\displaystyle \infty$ in $\displaystyle \mathbb{Z}_{\infty}$ - and so there are neither any zero divisors. I really have no idea about this question, however, and these are just my wildest guesses.

I really appreciate any insights. I am a total beginner at algebra.

2. 1. Here is my approach to it. Usually we have $\displaystyle \mathbb{Z}_n =$ set of integers modulo $\displaystyle n$ or $\displaystyle \mathbb{Z}_n =$ residue classes mod $\displaystyle n$. Either way this set contains all the integers mod $\displaystyle n$, which means this set consists of the remainders after division by $\displaystyle n$ of the integers. And we are choosing the smallest positive ones.

For $\displaystyle \mathbb{Z}_{\infty}$ we are dealing with infinity, so it can be tricky, but lets try to do the same thing. Infinity is usually defined as being something that is greater than any positive integer, that is, if $\displaystyle k \in \mathbb{Z}^+$, then $\displaystyle k < \infty$. So for any positive integer $\displaystyle k$ we already have that $\displaystyle k$ is smaller than the thing we are modding out by, so we have that $\displaystyle k$ is its own remainder. Thus $\displaystyle \mathbb{Z}^+ \subseteq \mathbb{Z}_{\infty}$

Now, what about the negative integers. It gets trickier For a positive integer $\displaystyle k$ we already have $\displaystyle k < \infty$. Multiplying by $\displaystyle -1$ we get $\displaystyle -k > -\infty$. We need some help from $\displaystyle \mathbb{Z}_n$ to understand what is going on. When we have $\displaystyle k < n$, we have $\displaystyle -k > -n$, so when we mod by $\displaystyle n$, the least positive remainder is $\displaystyle -k+n = n - k$. If we try to apply this to the the case with infinity, we must get that the least positive remainder is $\displaystyle - k +\infty = \infty \equiv 0 \pmod \infty$. So then we have $\displaystyle \mathbb{Z}_{\infty} =$ non-negative integers.

Now don't really take my word for it. It's just how I would think about it. If your book has not mentioned what $\displaystyle \mathbb{Z}_{\infty}$ means then it probably does not matter that much what you do as long as it's logical and coherent. If it's actually important in the framework of the book, then there must surely be a definition of what $\displaystyle \mathbb{Z}_{\infty}$ means somewhere in the book. If you haven't checked thoroughly, check now.

2. In this question you are asked specifically about $\displaystyle \mathbb{Z}$ so I think you really don't need to worry about $\displaystyle \mathbb{Z}_{\infty}$. Among the integers, there are no zero divisors since for any non-zero $\displaystyle a,b \in \mathbb{Z}$, you have $\displaystyle ab \neq 0$. Now, lets talk about the units. For an element $\displaystyle a \in \mathbb{Z}$ to be a unit, it must be invertible, that is there must exist an element $\displaystyle b$ such that $\displaystyle ab = 1$. Since we are talking about the integers, the only integers with this property are clearly 1 and -1 since if we consider anything greater than 1 in magnitude, we have that the product has magnitude greater than 1.

3. Thank you so much for explaining! It has taken me a little while to fully internalize it because I'm so new to this. The book doesn't seem to mention $\displaystyle \mathbb{Z}_{\infty}$ in any other context, so I guess it's not that important.

I really appreciate all the help!

4. No problem! Shout if you have any more questions!