The

characteristicof a ring (with unity, say) is the smallest positive number n such that 1+1+⋯+1⏟n 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 a≤b
- divisibility a∣b (i.e. ∃c.b=ac)

They are mostly compatible – *usually* when a∣b, it holds a≤b.

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 n⋅1=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 n⋅1=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:A→B, it must hold charB∣charA. 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 I⊆Z, 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.)

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