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Let $F$ be a set, $\forall a, b, c \in F$, with two binary operations, addition and multiplication, defined on $F$, i.e. $(F, +, \times )$. Then $F$ is a field iff the following axioms hold:

  1. Closure:

    1.1. Addition: $\forall a,b \in F, \exists a + b \in F$.

    1.2. Multiplication: $\forall a,b \in F, \exists a \times b \in F$.

  2. Associativity:

    2.1. Addition: $\forall a,b,c \in F, (a + b) + c = a + (b + c)$.

    2.2. Multiplication: $\forall a,b,c \in F, (a \times b)\times c=a \times (b \times c)$.

  3. Identity element:

    3.1. Addition: $\exists e_{Additive} \in F, s.t. \forall a \in F, a + e_{Additive} = e_{Additive} + a= a$.

    (Edit - a correction, see comments).

    3.2. Multiplication: $\exists e_{Multiplicative} \in F, s.t. \forall a \in F, a \times e_{Multiplicative} = e_{Multiplicative} \times a= a$.

    (Edit - a correction, see comments).

  4. Inverse element:

    4.1. Addition: $\exists b, e_{Additive} \in F, \forall a \in F, s.t. a + e_{Additive} = a, a + b = b + a = e_{Additive}$.

    (Edit - defined $e_{Additive}$)

    4.2. Multiplication: $\exists b, e_{Multiplicative} \in F, \forall a \in F$, s.t. $a \times e_{Multiplicative} = a, a \times b = b \times a = e_{Multiplicative}$.

    (Edit - defined $e_{Multiplicative}$)

  5. Commutativity:

    5.1. Addition: $\forall a, b \in F, a + b = b + a$.

    5.2. Multiplication: $\forall a, b \in F, a \times b = b \times a$.

  6. Distributivity: $\forall a, b, c \in F, a \times (b + c) = a \times b + a \times c = b \times a + c \times a = (b + c) \times a$.

  7. The additive and multiplicative identity elements are different: $\exists e_{Additive}, e_{Multiplicative} \in F, \forall a \in F$, s.t. $a + e_{Additive} = a, a \times e_{Multiplicative} = a, e_{Additive} \neq e_{Multiplicative}$.

    (Edit - axiom 7 added (hopefully correctly formulated), see comments and answers.)

I want to know if the definition above is correct, if not why not, and if my use of quantifiers is correct, if not why not to avoid false statements and the such. Now, I know fields and whatnot are defined on wikipedia and other sources, including here, over a million times but I want to use "my own" ordering of these axioms, if that makes any sense, and ask the following questions.

  1. Are all of these axioms detailed enough to define a field? E.g. In $(6)$ I didn't specify left or right distributivity but it is implied because of $(5.2)$, if I am not mistaken or is it even required to do so. Did I miss anything like that?

  2. Using the numbering of the axioms above, if they are stated correctly, which axioms must hold for a structure to be a ring? Now, I am not enrolled in Abstract Algebra or such so please don't dive in too deep if possible.

Any help much appreciated, thank you.

Dick Armstrong
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  • Ah yeah, I edited to fix that. I missed that by accident. Thank you xP – Dick Armstrong Jan 04 '20 at 06:44
  • Do you mean as followed: $\exists e_{Additive} \in F, s.t. \forall a \in F, a + e_{Additive}=a$? – Dick Armstrong Jan 04 '20 at 06:57
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    It's kind of a technicality but for a field we need $e_{additive} \neq e_{multiplicative}$ i.e. in standard notation $0\neq 1$. (This only excludes the zero ring which consists of the one element $0=1$, but for reasons.) – Torsten Schoeneberg Jan 04 '20 at 06:58
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    Finally (for now), without commutativity you would have to state left and right versions both for the identities and the inverses, i.e. your correct remark about distributivity applies to numbers 3 and 4 as well. – Torsten Schoeneberg Jan 04 '20 at 07:02
  • I.e. commutativity, in a sense, should "come" before the identities and inverses to make this "wholesome"? – Dick Armstrong Jan 04 '20 at 07:06
  • Zero ring: if $0=1$ then does that mean $0+1=0$? Maybe something like a watch but instead of 12 hours there is only 1 hour on the clock, and each time the arms move they just go in circle back to where they started? – Dick Armstrong Jan 04 '20 at 07:15
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    Regarding $0 \neq 1$, should I perhaps update $(3.2)$ to state $\exists e_{Multiplicative} \in F, s.t. e_{Multiplicative} \neq e_{Additive}, and \forall a \in F, a \times e_{Multiplicative} = a$? If I want to exclude the Zero ring – Dick Armstrong Jan 04 '20 at 07:22
  • As currently formulated, the elements $e_{\text{Additive}}$ and $e_{\text{Multiplicative}}$ in (4.) are undefined (c.q. free). You obviously want them to be the ones that exist according to (3.), but that's not what the axioms are saying right now. – Magdiragdag Jan 04 '20 at 08:31

2 Answers2

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More succinctly, a field is a commutative ring in which every nonzero element is a unit.

  • What do you mean by "every nonzero element is a unit"? I don't understand – Dick Armstrong Jan 04 '20 at 07:17
  • 0 of F is not unit, 0 does not have inverse. – Maja the bee Jan 04 '20 at 07:21
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    @DickArmstrong a unit in a commutative ring is an element that has an inverse for multiplication i.e. $x$ such that $\exists x': xx'= 1$. In a ring $0x=0$ and we know $ 0 \neq 1$ so $0$ can not be a unit. Being a field means this is the only exception. – Henno Brandsma Jan 04 '20 at 07:21
  • So, the set of real numbers can still be considered a ring because you said "every nonzero element is a unit" and didn't say anything about zero element being a unit in a commutative ring. Is this correct to assume? – Dick Armstrong Jan 04 '20 at 07:27
  • @chris Even with the notation "ring" = "ring with $1$", for the standard definition of field you should still need to exclude $F$ being the one-element ring. –  Jan 04 '20 at 07:34
  • @Gae You may be right. Still, I like this way of remembering what a field is. –  Jan 04 '20 at 07:45
  • The category $\bf{Field}$ is contained in the category $\bf{CRing}$. –  Jan 04 '20 at 08:11
  • This may be off, technically. I probably have to say:. it's the full subcategory whose objects are fields. –  Jan 04 '20 at 08:33
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To sum up comments:

To your question 1: Your definition is correct when amended as follows:

  • (crucial:) Add an axiom 7: $e_{add} \neq e_{mult}$.
  • (formal:) In no.4 (and the amended no.7), somehow make clear that $e_{add}$ resp. $e_{mult}$ refer to the elements defined in no.3.
  • (optional:): In nos. 3,4 and 6, amend a "left" and a "right" version. As long as axiom 5 is in place, this is of course redundant, but it will help to clarify an answer to your second question.

To your question 2: For a ring, leave out 3.2, 4.2, 5.2 and the amended no.7. For a unital ring (which however often is tacitly assumed even if people just write "ring"), leave out nos. 4.2, 5.2, and 7, but keep 3.2. (in a left and a right version). For a commutative ring (which is almost always understood to be unital as well, and we'll follow that), leave out nos. 4.2 and 7.

By the way, if you leave out just 5.2 but keep everything else (including no.7 and with left and right versions where applicable (although some of them might turn out to be redundant), you get what is called a division ring or skew field.

Torsten Schoeneberg
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