I was trying to prove the following statement:
"If an integral domain has a finite characteristic then the characteristic of the integral domain is a prime number"
This made me look at the definition of an integral domain, as given in the book, "Topics in Algebra " by I.N Herstein (2nd Edition) in Chapter -3 (Ring Theory) page number-126. The definition is as follows:
If $R$ is a commutative ring, then $a\neq 0\in R$ is said to be a zero-divisor if there exists a $b\neq 0\in R,$ such that $ab=0.$
A commutative ring is an integral domain if it has no zero-divisors.
Later in the book, on page number-129, the definitions of the characteristic of an integral domain are given as:
Definition: An integral domain $D$ is said to be of characteristic $0$ if the relation $ma = 0,$ where $a\neq 0$ is in $D,$ and where $m$ is an integer, can hold only if $m = 0.$
The ring of integers is thus of characteristic $0,$ as are other familiar rings such as the even integers or the rationals.
Definition: An integral domain $D$ is said to be of finite characteristic if there exists a positive integer $m$ such that $ma = 0$ for all $a\in D.$
If $D$ is of finite characteristic, then we define the characteristic of $D$ to be the smallest positive integer $p$ such that $pa = 0$ for all $a\in D.$
The problem is with a remark that followed these definitions above. The remark was:
It is not too hard to prove that if $D$ is of finite characteristic, then its characteristic is a prime number.
Acccording to the definition of an integral domain as given in the book, the ring $R=\{0\}$ i.e containing only the zero element is also an integral domain.
Now, we note that, $m.0=0$ for any integer $m.$ So, $R=\{ 0 \}$ is not of characteristic zero. But, $R$ has a finite characteristic. The smallest positive integer that $m$ can be, in this case is $1.$ But $1$ is not a prime number. So, the thing given in the remark is false, because, even $R$ is an integral domain with a finite characteristic, the characteristic of $R$ is not a prime, and neither is it a composite number.
But nearly every where, I have found the statement:
"If an integral domain has a finite characteristic then the characteristic of the integral domain is a prime number"
true in general. This is why, I think that some definition in the book might be misleading or better say incorrect such that it contradicts such well- established statement.
Any help regarding this issue will be highly appreciated.
Typically, an integral domain is defined to additionally require that $0 \neq 1$.
Another way to express this requirement is as follows. For all $n \in \mathbb{N}$, for all sequences $a_1, \ldots, a_n$, if the product $a_1 \cdots a_n = 0$ then there is some $i$ such that $a_i = 0$. The case $n = 0$ is precisely saying that $0 \neq 1$. For if we have $1 = 0$, that’s the same as saying the empty product is $0$. But there is no $a_i$ at all that could equal $0$.
If you drop the requirement $0 \neq 1$, the above Lemma about finite products only applies when $n \geq 1$.