In the book "Linear Algebra" by Georgi E. Shilov, Chapter I the exclusion of $0$ from the numbers considered is justified by a note stating:
Given two elements $N$ and $E$, say, we can construct a field by the rules $N+N=E, N+E=E, E+E=N, N\cdot N=N, N\cdot E=N, E\cdot E=E$. Then, in keeping with our notation, we should write $N=0, E=1$ and hence $2=1+1=0$. To exclude such number systems, we require that all natural field elements be nonzero.
What leads to the assumption that $E+E=N$?
The author is saying that we are in general allowed to define $E+E=N$, and this results in a structure that satisfies the definition of a field. The smallest positive integer $p$ such that $p\cdot 1=0$ is called the characteristic of the field, and in that case the "natural number" $p$ is equal to $0$.
For whatever reason, the author doesn't want this to happen. A field where no sum of $1$s is $0$ is said to be of characteristic $0$. Basically the author is only considering fields of characteristic $0$.