# Thread: Abstract problems - rings

1. ## Abstract problems - rings

Hi,

1. Let E denote the set of even integers. Prove that with the usual addidion, and with multiplication defined by m*n=(1/2)mn, E is a ring. Is there a unity?

2. Prove that aČ-bČ=(a+b)(a-b) for all a,b in a ring R iff R is commutative.

I'm new to rings and am unsure how to get going with these problems. Any help would be great.

Thanks.

2. Originally Posted by jackiemoon
Hi,

1. Let E denote the set of even integers. Prove that with the usual addidion, and with multiplication defined by m*n=(1/2)mn, E is a ring. Is there a unity?
well, check the following properties..
1. Is E abelian under addition? (well, this is true for all integers/)
1.5. Is the operation * well defined, i.e is a*b = (1/2)ab still an even integer?
2. Is the operation * associative? a*(b*c)=(a*b)*c
3. Is DPMA satisfied? a*(b+c) = a*b + a*c

Originally Posted by jackiemoon
2. Prove that aČ-bČ=(a+b)(a-b) for all a,b in a ring R iff R is commutative.

I'm new to rings and am unsure how to get going with these problems. Any help would be great.

Thanks.
first, $(a+b)(a-b) = (a+b)a - (a+b)b = a^2 + ba - ab -b^2 = a^2-b^2 \Longleftrightarrow ba-ab=0$...

3. Hey Kalagota,

Thanks for the reply. What does DPMA stand for?

4. Originally Posted by jackiemoon
Hey Kalagota,

Thanks for the reply. What does DPMA stand for?
oh, Distributive Property of Multiplication over Addition..