Non-abelian rings

In abstract algebra, we like to define various algebraic structures that generalize some very concrete object. For example, a field is defined using properties that enable addition, multiplication, subtraction, and division in a way analogous to the real or rational numbers, and a group is defined using properties that generalize composition of invertible functions.

Now, a ring, which generalizes the integers, is defined based on addition, subtraction, and multiplication (we can’t divide integers in general, since e.g. 1/2 is not an integer). More precisely, they’re defined by the following properties:

1. For all a and b, a + b = b + a
2. For all a, b, and c, a + (b + c) = (a + b) + c
3. There exists a number 0 such that a + 0 = a = 0 + a for all a
4. For each a, there exists an element -a such that a + (-a) = (-a) + a = 0
5. For all a, b, and c, a * (b * c) = (a * b) * c
6. For all a, b, and c, a * (b + c) = a * b + a * c = (b + c) * a

That a + b and a * b are always elements of the ring is implied in the definition. Some people require the ring to have an element 1 such that a * 1 = 1 * a = a for all a; the definition I learned doesn’t, and I think this way it works better; when the element 1 exist, I just call it a ring with 1. In similar vein, if for all a and b, a * b = b * a, the ring is called commutative.

Now, so far, so good. But what if we drop property 1? The ring will still form an additive group (groups are defined by properties 2-4 of rings; if they satisfy #1, they’re called abelian; why it is “abelian” and not “commutative” is beyond me), and multiplication will still distribute over addition.

However, Graeme Taylor of Modulo errors commented on Good Math, Bad Math, linking to a Usenet post showing that non-abelian rings are fairly restricted: in particular, every ring with 1 is abelian. This is because (a+a)+(b+b) = (1+1)(a+b) = (a+b)+(a+b) by distribution, so that a+b = b+a by cancellation.

9 Responses to Non-abelian rings

  1. I didn’t realize you were in mathematics. Were you at the Technion previously or some other university?

  2. Alon Levy says:

    No, I went to the National University of Singapore. I left Israel in the middle of 7th grade.

  3. I’ve been doing some more digging, and it seems unlikely that there are non-pathological examples of non-abelian rings. The only examples I’ve found are those given in the usenet post; mangling a group into a ring by adding a trivial multiplication, in which case the ‘ring theory’ that arises is unlikely to offer any more insights than the obvious choice of group theory in examining such an object.

    Based on discussion with a former lecturer of mine, it seems there are natural reasons for a ring to admit abelian addition, and thus the issue over axioms is more stylistic (whether to offer full abstraction, or a more accurate representation of the intended object). This is because algebra is essentially concerned with structure and structure preservation, and rings tend to emerge by considering the set of endomorphisms on a group with addition and composition as the pair of operations. Since there’s no point in looking at a structure preserving map without structure, the underlying structure tends to be an abelian group, which forces addition in the endomorphism ring to commute.

    Generalising, if one dabbles in true abstract nonsense, there are category theoretic reasons that lead to a necessarily abelian addition on the set of morphisms of a category under surprisingly minimal axioms. But that level of abstraction is beyond me!

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  8. Work Online…

    Although this has not been proven and maybe in the future it has the capacity to…

  9. Dejan Govc says:

    This algebraic structure is also known as a “distributive near-ring”. And there do exist non-trivial examples, in case you might still be interested. It seems people study structures like this quite seriously:

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