4.( 1) is a ring, an integral ring and a domain;

(2) Not a ring, because addition is not closed;

(3) It is a ring, not an integral ring and field, because multiplication has no unitary elements;

(4) It is not a ring, because the negative element of positive integer about addition does not exist and does not form a group about addition;

(5) Not a ring, because multiplication is not closed.

6.( 1)(-a)(-a)=-(a a)= 1，(-a)(-a)=-(a a)= 1

Therefore, -a is the inverse of (-a), and (-a) =-a is obtained according to the uniqueness of the inverse.

(2) (b a )(a b) = b (a a) b = 1，(ab) (b a ) = a (b b ) a = 1

So b a is the inverse of ab, and according to the uniqueness of the inverse, there is (a b) = b a