# Is the zero of a commutative ring not a zero divisor or is it “undefined?”

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?

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I wouldn't say that 0 of ring is "undefined", but instead something like "in definition 2 it is left open whether 0 is a zero divisor" or something like that. – coffeemath Oct 28 '12 at 9:39

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?".

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I was wondering, is it preferable to have a definition where zero isn't a zero divisor? If so, is there a particular reason why? – tcmtan Oct 28 '12 at 10:06
The definition of an integral domain is one case. It is defined to be a commutative ring, other than the trivial ring $\{0\}$, which has no zero divisors. So if one called $0$ a zero divisor, then there would be no integral domains! – coffeemath Oct 28 '12 at 11:01
No, the correct definition of an integral domain is that $0$ is the only zero divisor. For every non-trivial ring $0$ is a zero divisor (see my answer and arbitrary canonical algebra texts). – Martin Brandenburg Oct 28 '12 at 16:06
In "Basic Algebra" by Robert Ash, p 29, he says "If a and b are nonzero but ab=0, we say that a and b are zero divisors". So again Ash leaves open what to say about 0, as does Definition 2 above. And on the next page p 30 Ash defines an integral domain as a commutative ring with no zero divisors. I did find some other texts wherein for a module, 0 is considered a zero divisor; in that approach if the left (right) multiplication by $a$ is not injective, then $a$ was said to be a zero divisor. – coffeemath Oct 29 '12 at 0:19
Simple: 0 is not non-zero, so (by the first statement) it isn't a zero divisor (no need to apply, essentially). Case closed. – vonbrand May 21 '13 at 0:09

I see the reason for your scepticism, indeed, I already faced the same problem some time ago. At its heart, this problem arises from the lack of a well-defined semantics of a written sentence (which is the main reason why the mathematical formalism came up).

The intiution here is to add another sentence to the end of the definition:

"All other elements are not a zero-divisor."

Sometimes a similar sentence is even written out explicitly, otherwise it can be assumed implicitly.

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There are too many algebra texts which make assumptions in order to exclude pathological special cases, but in fact these assumptions are wrong! The correct definition of "zero divisor" has no "non-zero" in it!

An element of a commutative ring $r \in R$ is called regular if $r : R \to R$ is injective, i.e. $x \in R$ and $rx = 0$ implies $x=0$ (in the non-commutative case, one distinguishes between left- and right-regular). An element, which is not regular, is also called a zero divisor. In other words, $r \in R$ is a zero divisor iff there is some $x \in R$ such that $rx= 0$ and $x \neq 0$. In particular, $0 \in R$ is a zero divisor iff $R \neq \{0\}$.

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This is yet another definition of zero divisor, in terms of another term "regular". And this definition definitely as you say, applied to 0, if the ring is not the trivial ring, gives that 0 is a zero divisor. Seems like the old debate about whether 0 is a natural number or not... – coffeemath Oct 28 '12 at 9:45
This is not yet another definition, this is the correct definition. – Martin Brandenburg Oct 28 '12 at 12:10
OK but I did see some texts wherein zero divisor definition was not worded carefully enough so as to decide about 0. Another such, besides Ash, was Dummit and Foote's "Abstract Algebra". On page 228 in the Definition (only one on that page) "A nonzero element a of R is called a zero divisor if...", and in the (only) definition on page 229 he defines an integral domain as a commutative ring with $1 \ne 0$ without zero divisors. – coffeemath Oct 29 '12 at 7:35
Yes, this is adopted by many other texts too, whose authors are obviously uncertain themselves. – Martin Brandenburg Oct 29 '12 at 14:58
Guess it's one of those cases where since folks "know what they mean" by making statements involving a given term, they aren't careful about possible exceptions. – coffeemath Oct 29 '12 at 18:00