Why every field is a local ring, but the ring of integers $\mathbb{Z}$ not?

Question

It is said, that $\mathbb{Z}$ is trivially not a local ring, because the sum of any two non-units must be a non-unit in a local ring and for example $-2+3=1$. But why every field is said to be local ring? Isn't the violation of exact this rule not a violation for fields too (if we think for example on the field of rational numbers)?

Answer 1

Remember that an element $x$ of a ring is a unit if there is some $y$ such that $xy=yx=1$. In other words, the units are the elements with inverses wrt multiplication in the ring.

If $F$ is a field then an element $x$ of $F$ is a non-unit iff $x=0$. So the sum of any two non-units in $F$ is again a non-unit in $F$. So fields don't violate this characterization of local rings.

For example, in $\mathbb{Q}$, $2$ and $-3$ are units with inverses $1/2$ and $-1/3$ respectively.

Answer 2

https://en.wikipedia.org/wiki/Local_ring#Definition_and_first_consequences

A local ring is a ring which has aunique maximal ideal and if $p$ is prime the ideal generated by $p$ is a maximal of $\mathbb{Z}$.