# Freistetter's world of formulas: when is a ring a ring?

## When is a ring a ring?

The rings of math fit neither on fingers nor around planets - for they are completely abstract objects. But we use one of these rings every day.

Mathematics takes place in an abstract world of numbers, formulas and logic. Nevertheless, she often borrows the terms she uses to talk about this world from our everyday lives. So when mathematicians talk about "rings", in most cases they don't mean jewelry, but something that looks like this:

This isn't really a mathematical formula, but the formal expression of something called an algebraic structure. This can be described more clearly as a certain way of organizing elements. The R stands for a mathematical set of elements; the symbols + and · represent two combinations of the elements of this set (for example, but not necessarily, addition and multiplication).

So that the set and the two links form a "ring", certain conditions must be met. The set R must be a so-called abelian group with respect to the + combination. This means that it shouldn't matter whether you calculate a+b or b+a. The results of the arithmetic operations a+(b+c) and (a+b)+c must also be identical.

Also there must be a neutral element 0 in the set that does not affect the result of the join. It must therefore apply: a+0=a, and for every element in R there must be an inverse element which, when combined, produces the neutral element with it.

If the combination + actually denotes normal addition and the set R denotes the integers, then we all know the above definitions from school lessons. When adding whole numbers, it doesn't matter in which order it happens or which partial additions you do first. The number 0 is the neutral element, and for every integer there is an inverse element in the form of the same integer with a different sign.

## A ring of integers

As far as the second connection of a ring is concerned, the associative law must also apply here; Parentheses must therefore not play a role: (a b) c must produce the same result as a (b c). And here, too, there must be a neutral element. And finally, the distributive laws must also hold for all elements of the set R; a (b+c) must equal (a b)+(b c) and the result of (a+b) c must equal that of (a c)+(b c).

The integers, together with the usual addition and multiplication, form a ring, as you can easily see for yourself. Not the natural numbers, by the way, because there is no inverse element for the addition: the negative numbers are not part of the set of natural numbers.

Rings can also be much more abstract; instead of whole numbers, matrices or polynomials can represent the elements of R and + and · for corresponding arithmetic operations that are applied to these objects. By generalizing these concepts, however, the properties of such algebraic structures can be studied and understood much more comprehensively within the framework of "ring theory".

The arsenal of mathematical rings rivals any jeweler's shop: there are, for example, simple rings, local rings, integrity rings, Euclidean rings, half rings, almost rings, alternative rings, principal ideal rings, subrings, overrings, chain rings, semiperfect rings and even normal rings.

But which rings were exchanged privately between mathematicians on the recent Valentine's Day is not known.