Question

The following definition is needed for Exercises 41-43. Let R be a ring with identity. If there is a smallest positive integer n such that $\mathit{n}{\mathit{I}}_{\mathbf{R}}\mathbf{=}{\mathbf{0}}_{\mathbf{R}}$, then R is said to have characteristic n. If no such n exists, R is said to have characteristic zero.

Prove that a finite ring with identity has characteristic n for some $\mathit{n}\mathbf{>}\mathbf{0}$.

Short answer

It is proved that the finite ring with identity has characteristics n for some .n>0

Step-by-step solution

Definition of a ring with identity

A ring with identity would be ring R, which contains an element$1_R$ that satisfies the following axiom:

$ab=ba$for every $a\in R$. [Multiplicative identity]

Show that a finite ring with identity has characteristic n for some n>0

The set of a positive integer is represented by$Z_{+}$ . R is infinite and there exists $m,n\in Z$for$n>m$ in which$n{1}_R=m{1}_R$ indicates that $\left(n-m\right)1_R=0_R$for$n-m\in Z$ .

The characteristic of R could be less than $n-m$, however, this ensures that the characteristic is not zero.

Hence, it is proved that the finite ring with identity has characteristics n for some n>0 .