Determine if the indicated operations of addition and multiplication are defined on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether is has unity, and whether it is a field. 2ZXZ with addition and multiplication by components.

Question

Determine if the indicated operations of addition and multiplication are defined on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether is has unity, and whether it is a field.  2ZXZ with addition and multiplication by components.

terenzreignz · Answer

Still there?

anonymous · Answer

It should be right because ZXZ is a ring right?

terenzreignz · Answer

ZxZ is a ring, sure...
Since 2Z is a ring, then 2Z x Z should also be a ring...
that doesn't really do it, but you should arrive at that conclusion manually (IE testing out distribution and stuff)

terenzreignz · Answer

But yeah, 
$$\Large 2\mathbb{Z} \times \mathbb{Z}$$
is a ring... but is it a commutative ring?

anonymous · Answer

ok i will give it a shot on paper ：）thank you

terenzreignz · Answer

Wait, one quick question... does it have unity? :)

anonymous · Answer

well the question is written as this <2ZxZ,+,*> i am studying in a china so the question is in chinese.....so it says to prove its a ring by using the properites stuff

terenzreignz · Answer

Well, I suppose you can manually show that the set is closed under both component addition and multiplication.

anonymous · Answer

do you prove it for by using an element in 2z or just Z