Prove that the characteristic of a finite ring R is finite and it is a divisor of ∣R∣ where ∣R∣ denotes the number of elements of R.
Solution:
Let (R,+,⋅) is finite ring.
First Part
To prove, CharR is finite.
Since, by definition, the characteristic of a ring is the least positive integer n (if exists) such that n⋅α=0∀α∈R
where 0 is the zero element of R.
If no such positive integer n exists then CharR=0.
Therefore, in both case, the characteristic of a ring is finite whether the ring is finite or not.
Second Part
To prove, CharR is a divisor of ∣R∣.
Let ∣R∣=m and CharR=n
Then we need to prove that n∣m
Since (R,+,⋅) is ring then (R,+) is a abelian group.
Therefore, the Lagrange’s theorem m⋅α=0∀α∈R.
By division algorithm, ∃ two integers q and r such that
⟹⟹⟹⟹m=nq+rwhere0≤r<nm⋅α=(nq+r)⋅α∀α∈R0=n⋅(q⋅α)+r⋅α∀α∈R.n⋅0+r⋅α=0∀α∈R.r⋅α=0∀α∈R.
But r<nr \text{\textless} nr<n and nnn is the least positive integer such that n⋅α=0∀α∈R n\cdot\alpha=0 ~\forall~ \alpha\in \mathcal{R} n⋅α=0∀α∈R. Therefore the only possibility is r=0r =0r=0.