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The Power of the Distributive Law.

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Today, I learned about a very exciting statement when reading a thread in my favourite math forum.

Assume that you have an algebraic structure (A, +, \cdot) with two binary operations + and \cdot, where \cdot is distributive over +. Further assume that (a) there exists a neutral element 1 with respect to \cdot, and that (b) the operation + is associative, and that (c) the operation + is cancelable, i.e. for a, b, c \in A we have a + b = a + c \Rightarrow b = c and a + b = c + b \Rightarrow a = c. Note that (b) and (c) are satisfied for example if (A, +) forms a group.

These rather weak requirements already imply that + is commutative: if a, b \in A, then

a + b + a + b ={} & (a + b) \cdot 1 + (a + b) \cdot 1 = (a + b) \cdot (1 + 1) \\ {}={} & a \cdot (1 + 1) + b \cdot (1 + 1) = a + a + b + b.

Using the cancellation property, a + b + a + b = a + a + b + b implies b + a = a + b.

Using the aforementioned special case that (A, +) forms a group, we obtain:

Corollary.

Assume that the algebraic structure (A, +, \cdot) satisfies that (A, +) is a group and \cdot is distributive over + and has a neutral element. Then (A, +) is already an abelian group.

Therefore, if we loosen up the definition of a unitary ring by dropping the requirement that addition is commutative, the other axioms already force the commutativity of addition. Therefore, to get something more general than unitary rings (even if the multiplication is not associative or commutative), one has to make sure that a + b + a + b = a + a + b + b does not imply b + a = a + b, for example by asking for an addition not having the cancelation property.

Categories: Algebra
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Comments.

Jens wrote on March 29, 2012:

Hi Felix, I just came across your math blog incidentally and found (again) something interesting. :)

There is actually the notion of "near-ring", which is - besides semirings - another generalization of rings. Here, only a one-sided distributive law is postulated, and hence the addition might actually be noncommutative.

The endomorphisms of a group form such a near-ring, which is probably the most important example.

Felix Fontein wrote on July 11, 2012:

Thanks for the info, Jens. I didn't knew about "near-rings" yet. Also the example - endomorphisms of a group - is very helpful. I knew that they don't form a ring (unless the group is abelian, or maybe in some other random cases), but didn't knew the structure they form had a name.