Set of Integers not a field

Question

I read that the set of Integers $Z$ is not a field because it does not satisfy the Identity Axiom $X×X^{-1}=1$ The example given was that, according to the Identity Axiom, for a nonzero integer such as 2 there should exist a inverse $n$ such that $2n=1$, but that is impossible because 1 is an odd number.

I do understand what is meant by that explanation. However, when I thought about it a little I didn't understand why that is the case. The reason I don't quite understand it is illustrated by the fact that the number $3$ multiplied by its inverse $3^{-1}=1/3$ does equal $1$. So, if the inverse of $2$ is $2^{-1}=1/2$, shouldn't $2×1/2=2/2=1$, and thus satisfy the Identity Axiom?

Your problem is that $\frac{1}{2}$ does not belong to $\mathbb{Z}$. For a ring $R$ to be a field you need that every element in $R$ admits an inverse still in $R$. — Ender Wiggins, Mar 18 '17 at 15:37
"there should exist a inverse $n$ such" — correction: there should exist an inverse integer $n$ such — celtschk, Mar 18 '17 at 15:59

almosteverywhere · Answer 1 · 2017-03-18T16:01:27.390

11

In general, $n \in \mathbb{Z} \nRightarrow \frac{1}{n} \in \mathbb{Z}$.

edited Mar 18 '17 at 16:01

answered Mar 18 '17 at 15:42

almosteverywhere

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Set of Integers not a field

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