charesteristic

CHARACTERISTIC

Definition 1 – 17
Let R be a ring . If there exists a positive integer n such that
na = 0 for all a ? R , then the smallest positive integer with this
property is called the characteristic of the ring . If no such positive
integer exists , then R is said to be of characteristic zero ( that is ,
n = 0 is the only integer for which na = 0 for all a ? R ) . We shall
write char R for the characteristic of R .

Example 1
Find char Z5 ?
Solution
Z5 = { [0] , [1] , [2] , [3] , [4] }
5 [0] = [0] +5 [0] +5 [0] +5 [0] +5 [0] = [0]
5 [1] = [0] ?
5 [2] = 5 [3] = 5 [4] = [0] ?
Hence char Z5 = 5 .

Example 2
Find char P(X) ?
Solution
Let A ? P(X) ,
2A = A ? A = ( A – A ) U ( A – A ) = ? = 0
Hence char P(X) = 2 .


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