Question

how do you find the chararacteristic of Z6 of a ring? specifically Z6?

Answer 1

I assume you mean Z/6Z. Just consider the definition of characteristic. What is the least number of times you can add the identity of the ring, 1, to itself before you get zero? In this case, it is 6. In general the characteristic of Z/nZ is n. It can be a little trickier for rings which are products of Z/nZ's, like Z/2Z x Z/2Z. In this case, even though the ring has 4 elements, its characteristic is 2.