1. Axioms of Ring

1 Each of the following (with the usual operations of addition and multiplication) fails
to be a ring. In each case, you should prove this by showing one of the ring axioms
which is not satisfied.
(1) N, the set of natural numbers.
(2) 3Z+1, the set of odd integers.
(3) The set of invertible 2x2 real matrices.
(4) The set of polynomials in which the coefficient of x3 is zero.
(5) The set of vectors in real 3-dimensional space.

For (1): {1,2,3,4,5,.....} Identity law or zero law is not satified thus (1) is not a Ring. Am i correct?

(2): {.......,-5,-3,-1,1,-3,-5......} Identity law or zero law is not satified thus (2) is not a Ring. Am i correct?

I have no idea how to solve (3),(4),(5) any help would be greatly appreciated.

Thank you.

2. Originally Posted by charikaar
1 Each of the following (with the usual operations of addition and multiplication) fails
to be a ring. In each case, you should prove this by showing one of the ring axioms
which is not satisfied.
(1) N, the set of natural numbers.
(2) 3Z+1, the set of odd integers.
(3) The set of invertible 2x2 real matrices.
(4) The set of polynomials in which the coefficient of x3 is zero.
(5) The set of vectors in real 3-dimensional space.

For (1): {1,2,3,4,5,.....} Identity law or zero law is not satified thus (1) is not a Ring. Am i correct?

(2): {.......,-5,-3,-1,1,-3,-5......} Identity law or zero law is not satified thus (2) is not a Ring. Am i correct?

I have no idea how to solve (3),(4),(5) any help would be greatly appreciated.

Thank you.
(3) the zero matrix is not in the set.

(4) not closed under multiplication

(5) what operation is serving as product here? Cross product - no multiplicative identity.

RonL