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Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms:

  • Closure under addition. If $x, y \in F$, then $x = y = I$, so $x + y = I + I = I \in F$.
  • Closure under multiplication. $x \times y = I \times I = I \in F$
  • Existence of additive identity. $\forall x \in F$ (i.e., for $x=I$), $x + I = x$, so $I$ is the additive identity.
  • Existence of mulitiplicative identity. $\forall x \in F, x \times I = x$, so $I$ is the multiplicative identity.
  • Additive inverse. $\forall x \in F, \exists y = I \in F: x + y = I$
  • Multiplicative inverse. $\forall x \in F, \exists y = I \in F: x \times y = I$. However, because the additive identity need not have a multiplicative inverse, this is a vacuous truth.
  • Commutativity of addition. $\forall x, y \in F, x + y = I = y + x$
  • Commutativity of multiplication. $\forall x, y \in F, x \times y = I = y \times x$
  • Associativity of addition. $\forall x, y, z \in F$, $(x + y) + z = I + I = I$ and $x + (y + z) = I + I = I$, so $(x + y) + z = x + (y + z)$
  • Associativity of multiplication. $\forall x, y, z \in F$, $(x \times y) \times z = I \times I = I$ and $x \times (y \times z) = I \times I = I$, so $(x \times y) \times z = x \times (y \times z)$
  • Distributivity of multiplication over addition. $I \times (I + I) = I$ and $I \times I + I \times I = I$, so $\forall x,y,z \in F, x(y+z) = xy+xz$

Based on the above, $\{I\}$ seems to qualify as a field.

If $I$ is assumed to be a real number, then the unique solution of $I + I = I$ and $I \times I = I$ is, of course, $I = 0$.

So, is {0} a field, or is there generally considered to be an additional field axiom which would exclude it? Specifically, is it required for the multiplicative identity to be distinct from the additive identity?

Dan
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    This is not a field. All fields must contain at least two distinct elements, the additive identity and the multiplicative identity. – Potato Jun 22 '13 at 18:44
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    Often, one stipulates that $1 \neq 0$ in a field. The mathematical object you have specified doesn't really behave the way a field "should". For instance, one generally thinks of $0$ as a non-invertible element, and in fact this is the case for all fields with at least two elements. – Alexander Jun 22 '13 at 18:47
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    See http://ncatlab.org/nlab/show/too+simple+to+be+simple . The zero ring isn't a field for the same kind of reason that $1$ isn't prime. – Qiaochu Yuan Jun 22 '13 at 18:50
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    You may be interested in reading this. – Jay Jun 22 '13 at 21:36

6 Answers6

37

In the comments I claimed that the zero ring isn't a field for the same reason that $1$ isn't prime. Let me make this connection precise.

By the Artin-Wedderburn theorem, any semisimple commutative ring is uniquely the direct product of a finite collection of fields (with the usual definition of field). This theorem fails if you allow the zero ring to be a field, since the zero ring is the identity for the direct product: then for any field $F$ we would have $F \cong F \times 0$.

The "correct" replacement for the axiom that every nonzero element has an inverse is the axiom that there are exactly two ideals. The zero ring doesn't satisfy this axiom because it has one ideal. This has geometric meaning: roughly speaking it implies that the spectrum consists of a point, whereas the spectrum of the zero ring is empty.

Analogously, if you want every graph to be uniquely the disjoint union of connected graphs, you need to require that the empty graph is not connected. Here the "correct" replacement for the axiom that there is a path between every pair of vertices is the axiom that there is exactly one connected component.

Qiaochu Yuan
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  • This is illuminating! – Potato Jun 22 '13 at 18:59
  • But with that sort of thinking, one might just alter the A-W theorem to say "up to factors of ${0}$ as we say that factorizations are equivalent "up to multiplications by units" :) – rschwieb Jun 22 '13 at 19:02
  • @rschweib: units are irrelevant; take the corresponding unique factorization statement for ideals in Dedekind domains. – Qiaochu Yuan Jun 22 '13 at 19:08
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    @QiaochuYuan I am always puzzled by comments that begin with an oppositional tone and then continue by reiterating my point. The comment was meant to be lighthearted anyhow. It's not as if I was pointing out something wrong! Also, please use tab completion to spell my name right, so I don't miss comments. Thanks! – rschwieb Jun 22 '13 at 19:17
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    @Qiaochu: in that case one could say "up to factors of $R$". But I agree with what you're saying nevertheless. – Pete L. Clark Jun 22 '13 at 19:23
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    @rschwieb: but I'm not reiterating your point at all. The point is that you often runs into free monoids and there's a natural choice of free generators of these monoids. You want a term that picks out precisely these free generators and nothing else; in particular, you don't want it to pick out the identity. – Qiaochu Yuan Jun 22 '13 at 19:32
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    @QiaochuYuan To express what I meant with what you wrote: "copies of units ${0}$ are irrelevant". It looks like Pete sees what I'm thinking. If it's true that upon some super-deep categorical analysis you come up with that you can actually find a distinction, you are still wasting your time: the comment was just tongue-in cheek. – rschwieb Jun 22 '13 at 19:37
  • Hrm... seems I can't figure out strikethrough... it isn't ? – rschwieb Jun 22 '13 at 19:38
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    @rschwieb: comments don't support strikethrough. See http://meta.stackexchange.com/questions/31228/comments-should-support-the-strike-element . – Qiaochu Yuan Jun 22 '13 at 19:45
35

The short answer has already been given, that in general we do not want $1=0$. The element $I$ of the singleton set you are talking about is typically chosen to be $0$ instead, so that's why I'm using $\{0\}$ to denote the ring.

Here's one reason why the ring $\{0\}$ wouldn't make a good field:

$\{0\}^n\cong \{0\}$ for any $n$ as a "vector space," which doesn't fit very well with uniqueness of dimension for the vector spaces over other fields.

Although it is not about a bona-fide field, you might be interested in the wiki article on "field with one element". It gives motivation for an exceptional situation where it makes a little sense to think about something like a field with one element, but it isn't really a set satisfying the field axioms.

Again, the topic of the linked article in the previous paragraph is not actually a field at all, despite the name given to it. Seeing "field" and "one element" makes it tempting to conclude that $\{0\}$ is the subject of discussion (but it isn't).

Martin Sleziak
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rschwieb
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    "Field with one element" does not refer to the zero ring. It refers to something that isn't a ring at all. (Among other things, it refers to something that ought to be the initial object in a slightly enlarged category of commutative rings, but the zero ring is the terminal object.) – Qiaochu Yuan Jun 22 '13 at 18:50
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    It seems that the argument with $\left{0\right}^n \cong \left{0\right}$ is exactly why the definition of a field requires $1 \neq 0$. – Manos Jun 22 '13 at 18:54
  • @QiaochuYuan Ah, I think I remember this, I think I re-fell for the misconception that $F_1$ and the zero ring are the same. I guess that my further reading suggestion needs to be moved around so it doesn't support this misconception. – rschwieb Jun 22 '13 at 18:58
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    I should also add for other readers that the user is clearly talking about the ring ${0}$ and not about "the field with one element" at the wiki, so there is nothing wrong with me denoting it as ${0}$. The further reading suggestion is just informational. – rschwieb Jun 22 '13 at 19:05
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    I think you should remove the link entirely. It's misleading. The zero ring has nothing to do with the field with one element. – Qiaochu Yuan Jun 22 '13 at 19:06
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    @QiaochuYuan That suggestion is wrong because it is clearly a pedagogical blunder. As we've just seen, the two notions are easily confused. If I'm careful about saying $F_1$ isn't really a field (as I am being careful to) then the post is much better with the link than without. When teaching students, please do not avoid talking about commonly confused things... that fosters misconceptions! – rschwieb Jun 22 '13 at 19:13
  • That said, I hope anybody heeding my advice above is very careful to be clear when talking about commonly confused topics! That's a challenge in itself... – rschwieb Jun 22 '13 at 19:28
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    @rschwieb: then can you at least preface the link with a disclaimer like "be careful not to confuse the zero ring with..."? Right now the first sentence of the last paragraph seems more likely to promote rather than clear away confusion. It doesn't help that you never name the zero ring. – Qiaochu Yuan Jun 22 '13 at 19:37
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    This attacks the usefulness of an abstract notion of field that does not have distinct 0 and 1. If an abstraction is not useful, we don't usually bother with it. – ncmathsadist Jun 22 '13 at 21:54
  • All the answers so far are good. I'm giving this one the checkmark because it's the simplest explanation of why there's a $1 \ne 0$ requirement. – Dan Jun 23 '13 at 04:02
  • @QiaochuYuan I guess I can't argue about erring on the side of caution: prefaced and postscripted it with more danger signs. I didn't bother introducing "zero ring" because it is just more terms and because I can't supply non-confusing links to wikipedia about it. Apparently the wiki articles are set up to call your zero ring a "trivial ring" and rngs with zero multiplication "zero rings". Thank you for helping improve the answer. – rschwieb Jun 24 '13 at 10:55
  • @rschwieb: oh, hmm. I may have to change the terminology I use then. – Qiaochu Yuan Jun 24 '13 at 19:32
  • @rschwieb: Could I also argue that a ring $R(M, +, \cdot)$ must have two distinct operations $+$ and $\cdot$? In this case it would be obvious that every ring (and also every field) must have at least two elements. – Martin Thoma Jun 29 '13 at 08:20
  • @moose that would exclude the trivial ring ${ 0 }$ which is generally always considered a ring. – Ryan Sullivant Jul 09 '13 at 00:31
  • What do you mean by uniqueness of dimension? Is not dimension still unique? It is just that all the powers have the same dimension… – Vivaan Daga Feb 22 '23 at 15:51
  • @Shinrin-Yoku Yes, that's just it. The dimension is no longer linked to the power, as it is in the normal case. – rschwieb Feb 22 '23 at 17:28
19

Let me compile some comments into an answer.

By now it seems clear that the question is predicated on a misstatement / faulty memory of the field axioms. I claim that it is completely standard that $1 \neq 0$ is included in the field axioms. By "completely standard" I mean that every reputable source on the subject in the last $50$ years does it this way, and that at least 99.9% of people who speak about "fields" in the abstract algebraic sense mean to include this axiom.

If you don't include this axiom, then, as others have pointed out, the only additional structure you get is what is usually referred to as the zero ring: the unique -- up to isomorphism -- ring with a single element, which therefore acts as both an additive identity $0$ and a multiplicative identity $1$.

(The zero ring is certainly a ring, although a small minority of reputable sources exclude it. They shouldn't do so: we want, for any ideal $I$ in a ring $R$, to define the quotient ring $R/I$. If $I = R$ then this quotient is the zero ring. You might think "Well, fine, we'll just say that we can't take the quotient by the ideal $I = R$. For instance, maybe we'll say that $R$ is not an ideal of $R$ at all." But this is a bad convention, especially the latter, since in practice an ideal is often given by a set of generators, and usually one cannot easily tell by looking at the generators whether they generate the unit ideal $R$ or not. For instance, if $a_1,\ldots,a_n$ are integers, then according to this convention we can only consider the ring $\mathbb{Z}/\langle a_1,\ldots,a_n \rangle$ if there is some $d \geq 2$ which divides each of $a_1,\ldots,a_n$. So we have work to do before we know whether we can write down the quotient ring or not! In more complicated rings this becomes more difficult or practically impossible.)

The reasons for excluding the zero ring from being a field have already been well-described by others. In particular the modern perspective on commutative rings is to focus attention on modules over the ring. (As an aside, the Morita theory in the commutative case simply says that two commutative rings with isomorphic module categories are isomorphic.) But the only module over the zero ring is the zero module. So unlike all other rings satisfying the remaining field axioms, there are no nontrivial vector spaces over the zero ring and hence no nontrivial linear algebra. We also have the important notion of a simple module, which is a nonzero module with only the zero module as a submodule, and there is the important fact that if $N \subset M$ are $R$-modules, then $N$ is a maximal submodule of $M$ iff $M/N$ is a simple module.

Then there is the notion of a simple commutative ring, which is a ring such that $R$ is simple as an $R$-module. This means that $R$ is not the zero ring but the only ideals of $R$ are $(0)$ and $R$. It is easy to see that a commutative ring if simple iff it is a field. In particular, since ideals are $R$-submodules of $R$, an ideal $I$ of $R$ is maximal iff $R/I$ is a field. As Qiaochu Yuan likes to say, the zero ring is "too simple to be simple": including it as a field would screw all this up.

Let me end with some comments about forgetting about the field axiom $1 \neq 0$. In the context of analysis courses, fields appear at the beginning but only in a very shallow way. Every analysis text I know which discusses fields includes $1 \neq 0$ among the field axioms. However, I just checked Rudin's Principles of Mathematical Analysis and Spivak's Calculus, and though they include it, they do so in a way which makes it easy to forget. Namely they put it in as a sort of rider in the axiom about the existence of a multiplicative identity. From the perspective of abstract algebra this is a strange approach, since the existence of a multiplicative identity is one of the ring axioms, and there is a ring in which $0 = 1$, so we shouldn't be saying that $0 \neq 1$ in the same breath as we are averring the existence of $1$. But you don't encounter the definition of a ring in real analysis, typically.

On the other hand, notice how the field axioms are stated in $\S$ 1.2 of these honors calculus notes written by an algebraist. It makes it harder to forget that $0 \neq 1$!

Martin Sleziak
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Pete L. Clark
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    "Linear algebra over the zero ring is trivial and degenerate" is very different from "we cannot do linear algebra over the zero ring". –  Jun 22 '13 at 20:40
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    @Hurkyl: Really, "very different"? I'm not sure I agree -- if all vector spaces are trivial, then what is there to do? -- but I changed the language anyway. – Pete L. Clark Jun 22 '13 at 20:52
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    Although $0\neq1!$, one does have $1=0!$. – Marc van Leeuwen Jun 24 '13 at 11:55
  • @Hurkyl: The real problem of doing linear algebra over the zero ring is not that it is trivial and degenerate, but that there are statements that are true over fields but false for modules over the zero ring; this would force making exceptions for the zero ring. One such statement is that if a vector space has two ordered bases, these bases have the same cardinal: every sequence of elements of the (trivial) module is a basis, because linear dependence is inexistent (it requires a non-trivial relation). Even without order: both the empty set and the singleton zero vector are bases. – Marc van Leeuwen Jun 24 '13 at 12:08
  • @Marc: I think both are problems. But I agree with what you go on to say. As I commented above, it is very likely that allowing the zero field in linear algebra will actually lead to false statements of linear algebra theorems (unless you have enough care to put "except for the zero field" in many theorem statements, but surely anyone who did this would realize that the zero field is simply not desired after all). – Pete L. Clark Jun 24 '13 at 17:03
  • -1 for clapping to yourself. "Loses the fruit of his good action, those who price it by their own mouth." Confucius. –  Jan 07 '14 at 16:27
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    @Karene: I have written hundreds of pages of lecture notes for honors calculus and made them freely available on the web. The main drawback to having free, online only notes is that if people don't know about them, they can't read them. So I refer to them in my answers on this site and elsewhere. Calling that behavior "clapping to yourself" seems ungracious. Are you really suggesting that by calling attention to my volunteer work in a slightly positive way I am undoing all the good of that work? That's discouraging. – Pete L. Clark Jan 07 '14 at 22:06
  • @PeteL.Clark Writing the notes is the good action. The "On the other hand" after criticizing the exposition in one of the best written books in analysis is what sounds out of place. –  Jan 07 '14 at 23:16
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    @Karene: And to be clear, you think that being critical of exposition in other (not freely available) texts while liking the exposition in my own (freely available) text is enough to "lose the fruit of the good action"? Hundreds of hours of work are vitiated by a few lines of criticism, with a negative net outcome? (This is the message that your downvote conveys.) And if one is going to (very mildly) criticize exposition in standard texts, is it better to be purely negative or to offer up an alternative? – Pete L. Clark Jan 08 '14 at 02:14
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One requires $1_{F}\neq0_{F}$ so $\{0\}$ is not a field.

Belgi
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    "Usually" doesn't seem strong enough. I claim that $1 \neq 0$ is always one of the field axioms; even commutativity of multiplication is more negotiable (especially among older texts written in French). Can you supply a single example of a source which omits $1 \neq 0$ from the field axioms? – Pete L. Clark Jun 22 '13 at 19:20
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    @PeteL.Clark - I have edited it to remove the word "Usually". I think that it my introduction to rings course the zero ring was considered a field, but I'm not sure. Thanks commenting about this! – Belgi Jun 22 '13 at 19:23
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    "I think that it my introduction to rings course the zero ring was considered a field, but I'm not sure." This seems very unlikely, as the theory simply doesn't work as it should if you do this, unless at every stage you add provisos to take it out. For instance the basic fact: "An ideal $I$ in a commutative ring $R$ is maximal iff $R/I$ is a field." would be thoroughly ruined. – Pete L. Clark Jun 22 '13 at 19:27
  • This is not the first time someone has told me "I think I was told in a course X", where X is something highly awkward at best. Whenever this can be corroborated with an actual written source it turns out their memory was mistaken. – Pete L. Clark Jun 22 '13 at 19:35
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    It happens to me too that there are things I believed/remembered being told in a class that I was not in fact told. At any point in my graduate career if you had asked me whether a finite degree field extension could have infinitely many intermediate extensions, I would have said of course not, as one learns in undergraduate algebra. But this is wrong and -- therefore! -- was not taught to me in undergraduate algebra. I think the moral is not to say "I was taught this in class" as an argument for mathematical convention or truth. – Pete L. Clark Jun 22 '13 at 19:36
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    I would be surprised if most introductory courses mentioned the zero ring at all. I certainly don't recall it ever being mentioned in my courses. – Qiaochu Yuan Jun 22 '13 at 19:41
  • I don't recall my Elementary Real Analysis course ever explicitly mentioning the $1 \ne 0$ requirement (If I had, I wouldn't have asked the question), but it has been a while. – Dan Jun 22 '13 at 19:44
  • @Dan: I think that's because many people prefer not to think about the zero ring at all. – Qiaochu Yuan Jun 22 '13 at 19:52
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    @Qiaochu, Dan: notice that both of you are now relying on memories of courses taken years ago. As I said above, such memories are often unreliable. By way of comparison I just checked Rudin's Principles, Spivak's Calculus and the wikipedia article http://en.wikipedia.org/wiki/Field_%28mathematics%29 and they both include $1 \neq 0$ as one of the field axioms. – Pete L. Clark Jun 22 '13 at 20:00
  • @PeteL.Clark - I have checked: In my linear algebra course we did not mention that $1\neq0$ in the definition of a field.

    In my introduction to rings we have defined a field to be a commutative ring with unity and division, again not mentioning $1\neq0$

     – Belgi Jun 22 '13 at 20:05
  • @Belgi: Do you mean that you used a text in which $1 \neq 0$ does not appear as one of the field axioms? If so, please do tell me which one. (If you don't mean the course text, I wonder how you can "check" whether something was ever mentioned over the course of several months of lectures? Or do you mean that you asked the instructor?) – Pete L. Clark Jun 22 '13 at 20:40
  • And if so, what does your text say about the dimension of the vector space $F^n$? – Pete L. Clark Jun 22 '13 at 20:45
  • @PeteL.Clark - In linear algebra it was easy, there are the original video lectures and many notes from that class. From introduction to rings I have my own notebook scanned, as well as a definitions documents I made during that class. I never saw any textbook which didn't add this detail, so I revised my answer accordingaly – Belgi Jun 22 '13 at 20:46
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    @Belgi: I guess what you're saying is that the correct definition of a field was not given in the lectures of your linear algebra course. That's believable. Many times instructors give in-class presentations which are not 100% "formally correct" for pedagogical reasons -- and sometimes, unfortunately, they make mistakes. That's why it's important for a course to have a reputable text. I apologize for doubting your memory, but most people (especially, older people like me) can't check theirs against videotaped lectures. – Pete L. Clark Jun 22 '13 at 21:03
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    @PeteL.Clark - I have a bad memory, believe me that you have a very good reason to doubt it :P – Belgi Jun 22 '13 at 21:48
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    @Pete: who is relying on a memory? I was offering an alternate hypothesis for why a course wouldn't mention the $1 \neq 0$ axiom. – Qiaochu Yuan Jun 22 '13 at 23:16
  • @Qiaochu: I have probably stated things too strongly. I apologize. – Pete L. Clark Jun 23 '13 at 00:29
  • @PeteL.Clark Damiano and Little, Linear Algebra. They don't include (probably by ignorance) the requirement $1\neq0$. Still the rest of the books seems to work fine without it. One thing is understanding why $1\neq0$ is almost always required for a field, another is qualifying people that don't as being wrong. Mathematics still works as fine as always if $1=0$ in a field, if $1$ is a prime, ... you just need to add a few points to the statements of some theorems. –  Jan 07 '14 at 16:14
  • @Karene: Yes, a definition can never be "wrong"; at worst it can have unpleasant and unintended consequences. You can do linear algebra without $1 \neq 0$ by excluding the case of $1 \neq 0$ in certain theorems, mainly the key one which says $\operatorname{\dim} F^n = n$. I don't think I said that people you omit this axiom are "wrong". I said that it would lead to wrong theorems unless you take certain steps to stick in extra hypotheses, as you've said. But from a pedagogical perspective there seems to be only disadvantages to allowing $1 = 0$, which explains why almost no one does so. – Pete L. Clark Jan 07 '14 at 22:27
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I am quoting Lang's Algebra p. 84: "A ring $A$ such that $1 \neq 0$ and such that every nonzero element is invertible is called a division ring...A commutative division ring is called a field." Hence by definition, a field must contain at least two distinct elements.

Martin Sleziak
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Manos
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A set $\mathbf{(F,+,\times)}$ is called a 'field' if $\mathbf{(F,+)}$ is an abelian group and $\mathbf{(F\setminus\lbrace e \rbrace,\times)}$ is an abelian group where $\mathbf{e}$ is the identity element of $\mathbf{(F,+)}$ , such that $(a + b ) \times c = a \times c + b \times c $.

This clearly demands the existence of at least two elements and something more.

user2902293
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