In the Contemporary Abstract Algebra book by Gallian it defines zero-divisors as follows:
Definition 1) A zero-divisor is a nonzero element $a$ of a commutative ring $R$ such that there is a nonzero element $b$ in R with $ab=0.$
In another coursebook it defines zero-divisors slightly differently.
Definition 2) A nonzero element $a$ of a commutative ring $R$ is a zero-divisor if there exists a nonzero element $b$ such that $ab=0$
Now it seems that they are equivalent, but there seems to me a slight subtlety in the ordering of wording.
In definition 1) it follows that zero is not a zero divisor since it fails to be nonzero to begin with. (Since we have not restricted the elements of $R$ in the definition)
In definition 2) however, it seems as if the zero of the ring is "undefined" (i.e $a$ is neither a zero divisor nor is it not a zero divisor) because definition 2) starts as "A nonzero element $a$ in……" and by beginning the sentence in this manner it seems to me that we have restricted the set in question to nonzero elements. Hence this definition only applies to nonzero elements so zero is undefined under this definition. So in other words for definition 2), if you wish to find an element that is not a zero divisor then you need to find a nonzero element $a\in R$ such that $(\forall b\in R) (b=0 \vee ab\neq0)$.
Is the differing interpretations a failure of my understanding of the sentence structure of the two definitions or does the order in this particular case really matter?
Best Answer
The two definitions are in spirit the same. However I agree that there is a slight difference. I'll comment on each...
Definition 1) A zero-divisor is a nonzero element a of a commutative ring R such that there is a nonzero element b in R with ab=0.
In this definition the term "zero divisor" is defined in such a way that one may apply it to any element $a$ of the ring $R$, in particular to 0. On applying it to 0 the first part of the definition immediately says 0 is not a zero divisor, by the phrase "is a nonzero element $a$ ..." What follows after this phrase is irrelevant to deciding whether 0 is a zero divisor, since we already know it is not one after the first part of the definition is read. In my opinion this (Definition 1) is the clearer of the two.
Definition 2) A nonzero element a of a commutative ring R is a zero-divisor if there exists a nonzero element b such that ab=0
In this definition, technically nothing is said about applying the definition to 0. That is, the "scope" of Definition 2 is the collection of all nonzero elements of R. If one tries to apply Definition 2 with $a=0$, the first part of the definition immediately bars the way of deciding whether 0 is a zero divisor, since the definition only speaks about nonzero elements. So in my opinion definition 2 does not give the answer to the question "is $0$ to be called a zero divisor?".