# Abstract problems - rings

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• Mar 1st 2009, 02:22 AM
jackiemoon
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.
• Mar 1st 2009, 04:00 AM
kalagota
Quote:

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

Quote:

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, \$\displaystyle (a+b)(a-b) = (a+b)a - (a+b)b = a^2 + ba - ab -b^2 = a^2-b^2 \Longleftrightarrow ba-ab=0\$...
• Mar 1st 2009, 06:03 AM
jackiemoon
Hey Kalagota,

Thanks for the reply. What does DPMA stand for?
• Mar 1st 2009, 08:19 PM
kalagota
Quote:

Originally Posted by jackiemoon
Hey Kalagota,

Thanks for the reply. What does DPMA stand for?

oh, Distributive Property of Multiplication over Addition..