Question

what is the characteristics of the given ring:
Z6 x Z15

Answer 1

I believe when you have a ring with unity, to find the characteristic, it suffices to check the "1" of the ring. You need to find the smallest positive integer n such that:\[n\cdot 1 = 0\]

Answer 2

The element that plays the role of "1" in this ring is (1,1), so basically you need to find the additive order of the element (1,1). What is the smallest positive n such that:\[n\cdot (1,1) = (0,0)\]

Answer 3

means,for ring with unity,we just find n such that n.1=0..then it is true that n.a=0 for all a in that ring..

Answer 4

right, because if n*1 = 0, then for any element x in the ring:\[n\cdot x = x+x+x+\cdots +x=x(1+1+1\cdots +1)=x\cdot(n\cdot 1)=x\cdot 0 = 0\]

Answer 5

what ring such that it has unity but \[n.1\neq0\],thus it has characteristic 0?

Answer 6

right. if there is no positive n such that n*1 = 0, then it has characteristic 0. Like the integers.

Answer 7

for 2Z,it has no unity right?means we have to check every element a in 2Z such that n.a=0...

Answer 8

That is correct, 2Z has no unity. However, it is a subring of the integers, which has characteristic 0.