Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number n such that 1+1++1n times=0, provided such an n exists. Otherwise, we define it to be 0.

But why characteristic zero? Why do we not define it to be instead? Under this alternative definition, the characteristic of a ring is simply the “order” of the additive cyclic group generated by the unit element 1.

My feeling is that there is a precise and convincing explanation for the common convention, but none comes to mind. I couldn’t find the answer in the Wikipedia article either.

Answer

There are two orderings of the set N={0,1,}:

  • magnitude ab
  • divisibility ab (i.e. c.b=ac)

They are mostly compatible – usually when ab, it holds ab.

Some definitions are phrased using “greater than” ordering, while in fact the “divisibility” ordering is the real essence.

For example, the greatest common divisor of a and b might be defined as the greatest number which is a common divisor of both a and b. Characteristic of a ring R might be defined as smallest number n>0 which satisfies n1=0.

Under such commonly taught definitions, it seems natural that gcd(0,0)= and charZ=.

However, those definitions implicitly rely on ideals, and are better phrased using divisibility order. The incompatibility is then more visible: 0 is the largest element in divisibility order, while it is smallest in magnitude order. Magnitude has no largest element, and often is added to cover this case.

So let’s formulate the definitions again, but this time using divisibility ordering.

  • The greatest common divisor of two numbers a,b is greatest number (in sense of ) that is a divisor of a and b (i.e. is smaller than a and b in divisibility ordering). This is prettier – gcd is now the operator in lattice (N,); it also forms a monoid, with 0 as identity element. Additionally, the definition can be adapted to any ring.
  • The characteristic of a ring R is the smallest number n (in sense of ) that satisfies n1=0. As a bonus, compared to previous definition, we can remove the n>0 restriction: zero is always a valid “annihilator” but it is often not the smallest one. Now we get charZ=0.

Characteristic is a “multiplicative” notion, like gcd. If you have a homomorphism of rings f:AB, it must hold charBcharA. For example, you cannot map Z2 to Z4 – in a sense, Z2 is “smaller” than Z4. “Bigger” rings have “more divisible” characteristic, their characteristics are greater in the sense of divisibility. And the “most divisible” number is 0. Another example is charA×B=lcm(charA,charB).

In a bit more abstract language: given any ideal IZ, we associate to it the smallest nonnegative element, under the divisibility order. By properties of Z, every other element of I is a multiple of it. Let’s call this number min(I).

We can now define gcd(a,b)=min((a)+(b)), and charR=min, where f \colon \mathbb Z \to R is the canonical map.

The definition of \operatorname{min}(I) works for any PID, it does not require magnitude order. In any PID, I = (\operatorname{min}(I)).

(I dislike saying the ideal \{0\} is “generated” by 0; although this is true, it also generated by empty set. We do not say that (2) is generated by 0 and 2.)

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Source : Link , Question Author : Srivatsan , Answer Author : sdcvvc

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