Characteristics of integral domain is either zero or prime || Ring theoryII HINDI

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What is Characteristic of a Ring
Definition Characteristic of a Ring
The characteristic of a ring R is the least positive integer n such that
nx=0 for all x in R. If no such integer exists, we say that R has characteristic 0. The characteristic of R is denoted by char R.
Thus, the ring of integers has characteristic 0, and Zn has characteristic n. .

Characteristic of a Ring with Unity
Let R be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of R is 0. If 1 has order n under addition, then the characteristic of R is n
proof
If 1 has infinite order, then there is no positive integer n such
that n . 1= 0, so R has characteristic 0. Now suppose that 1 has additive order n. Then n . 1 = 0, and n is the least positive integer with this
property. So, for any x in R, we have
n . x = x + x + .............. + 1 x (n summands)
=1x+ 1x+ 1x+ .............+1x (n summands)
=(1+1+............+ 1)x (n summands)
=(n . 1)x =0x =0.
Thus, R has characteristic n. Thus, R has characteristic n.
In the case of an integral domain, the possibilities for the characteristic are severely limited.

The characteristic of an integral domain is 0 or prime.
PROOF By Theorem 13.3, it suffices to show that if the additive order
of 1 is finite, it must be prime. Suppose that 1 has order n and that n =st,

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