Algebra, esp. ring theory is about the relationship between addition and multiplication. One concept of exceptional importance is the relation of being a $k$th root of a number (with $k\in \mathbb{N}$ not considered as an element of the ring):

$x$ is a $k$th root of $y$ when $\underbrace{x\cdot x\cdot \dots \cdot x}_{k \text{ times}} = y$.

An example of a proposition concerning roots is this:

In $\mathbb{C}$ the sum of all $k$th roots of $1$ is $0$.

Now I wonder why another concept – the concept of being part of a number, which can be defined in perfect analogy to roots – has not been found worth to get a name for its own and correspondingly is not found explicitly in any relevant theorem or proof (as far as I can see):

$x$ is a $k$th part of $y$ when $\underbrace{x + x + \dots + x}_{k \text{ times}} = y$.

As we write $x^k$ for $\underbrace{x\cdot x\cdot \dots \cdot x}_{k \text{ times}}$ we may write $k\times x$ for $\underbrace{x + x + \dots + x}_{k \text{ times}}$, related to but not to be confused with $k \cdot x$ when $k$ is an element of the ring.

Compare the definitions of being root, part, and divisior:

$x$ is a root of $y$ when there is a $k$ with $x^k = y$.

$x$ is a part of $y$ when there is a $k$ with $k \times x = y$.

$x$ is a divisor of $y$ when there is a $z$ with $z \cdot x = y$.

Note, that the concept of part somehow builds a bridge between roots and divisors.

The concept of being part of a number has one great appearance: In the definition of the characteristic of a ring.

The characteristic of a ring is the smallest number $k$ such that $1$ is a $k$th part of $0$.

But after this appearance the concept of a part steps back behind the curtain and seems not to be needed anymore.

My questions:

  1. Do I oversee something and an equivalent concept of being-part-of is important per se and used in propositions and proofs of ring theory and algebra (but in disguise and under another name)?

  2. If not so: How can this be understood? Why is the concept not so important per se?

If you find these questions unclear and too unspecific, maybe you can answer this one:

In which rings is being part and being divisor equivalent, i.e. $x$ is part of $y$ iff $x$ is divisor of $y$?

  • $\begingroup$ Derp, I'm an idiot. $\endgroup$ – Noah Schweber Oct 7 '18 at 13:10
  • $\begingroup$ Does this mean there are unital rings where part-of and divisor-of are not equivalent? An example would be great! $\endgroup$ – Hans-Peter Stricker Oct 7 '18 at 13:18
  • $\begingroup$ E.g. in $\mathbb{Q}$, $2$ is a divisor but not a part of $1$. (And sure.) $\endgroup$ – Noah Schweber Oct 7 '18 at 13:24
  • $\begingroup$ Obviously! I should have come up with this by myself. So the question remains open: How can the (unital or not) rings be characterized in which part-of and divisor-of are equivalent? $\endgroup$ – Hans-Peter Stricker Oct 7 '18 at 13:28
  • $\begingroup$ A necessary condition might be that the characteristic is $\neq 0$? Does it suffice? $\endgroup$ – Hans-Peter Stricker Oct 7 '18 at 13:39

I don't know much abstract algebra, so I apologize beforehand if this is too basic or uninteresting.

I think this concept may be related to torsion of $\mathbb{Z}$ modules: just as it is important to know when a ring has zero divisors, it is important to know if an abelian group has torsion, i.e. if it has elements that are 'a part of zero'. Specially since that gives information about the classification of the group in terms of the structure theorem.

As for the last question, if I am understaing correctly, the notion of being a part of can be understood when $k$ is an integer or a finite sum of the ring's unity, in which case we have that

$$ kx = (1+ \dots + 1)x = x + \dots + x = k \times x $$

for any $k,x$.

  • 1
    $\begingroup$ Note that this leaves open the possibility of something interesting happening with non-unital rings. $\endgroup$ – Noah Schweber Oct 7 '18 at 12:09

To be honest, I don't like the term part you introduced. First, "part" is an overused word in mathematics (think of the real part of a complex number, or the integral and fractional parts of a real). Moreover, root and part are basically the same notions, the only difference being the multiplicative versus the additive notation. Also note they are not ring notions, but semigroup notions. Thus it would be more natural, in my opinion, to call a part an additive root if you really need to introduce a new term.

Your notion of a divisor is confusing since you are using the letter $k$, which is supposed to represent a natural number. In a commutative monoid $M$, $x$ is a divisor of $y$ if there there exists a $z \in M$ such that $zx = y$, but $kx$ does not make sense if $k$ is a natural number.

  • $\begingroup$ When I asked this question some weeks ago I was not aware of Euclid's definitions V.I and VII.3 which both define "part" in a similar way than I do. $\endgroup$ – Hans-Peter Stricker Oct 29 '18 at 11:58
  • $\begingroup$ Concerning the use of the variable $k$ in the definition of a divisor: you are right, that's confusing. I changed it. $\endgroup$ – Hans-Peter Stricker Oct 29 '18 at 12:00

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