# Direct sums and prime numbers

In my linear algebra class today, I was reminded of an episode of Code Lyoko, where Aelita gives a rather odd definition for a pair of prime numbers.

http://youtu.be/wvwSaKaz4is?t=9m41s

A pair of prime numbers occurs when their highest common denominator is one. In a principal ring, it's equivalent to the fact that their direct sum is equal to one, but of course, with a non principal factorial ring, that is not the case.

What does this even mean? Can I perform a direct sum on two integers yielding another integer that is different from regular addition?

• I think it refers to direct sum of prime ideals. In a PID distinct primes are comaximal meaning $P+Q=R$ which is the same is there exists $p+q=1$ for $p\in P$ and $q\in Q$. Furthermore since a the ideal structure of a PID is in bijective correspondence of the ring, people often refer to prime ideals as just primes. – TheNumber23 Sep 20 '13 at 5:18
• At best that is not the definition of a pair of prime numbers, but of a pair of relatively prime numbers, which is a quite different condition (and which cannot be used to define prime numbers). Also one does not take a direct sum of numbers, but maybe of prime ideals or such (hard to guess from the citation). So it would be best to just forget that nonsensical phrase. – Marc van Leeuwen Sep 20 '13 at 5:25

There is a vague allusion to the fact that in a factorial ring (or more generally a gcd ring) two elements $a,b$ are relatively prime (or coprime; by definition this means thay have no non-invertible common divisors) if and only if their greatest common divisor is $1$, and that (only) in a principal ideal domain this is equivalent to being able to write $1$ as $1=sa+tb$, a sum of multiples of $a~$and$~b$. "Highest common denominator" is nonsense, born of confusion between "highest common factor" (a synonym for gretest common divisor) and "lowest common denominator".
It takes some effort to find any link with direct sums at all. It is a fact however that in modules over a principal ideal domain, the submodule of elements with $ab$-torsion (those becoming zero when multiplied by $ab$) can be decomposed as a direct sum of the submodules of elements with $a$-torsion and with $b$-torsion if $a$ and $b$ are relatively prime, and that a Bezout relation $1=sa+tb$ is instrumental in proving this.