If in Ring, evaluate (a+b)(c+d)

[apalmer3]

Homework Statement

If a,b,c,d $$\in$$ R, evaluate (a+b)(c+d). (R is a ring.

Homework Equations

The Attempt at a Solution

I think that it's simple foiling, but I'm not sure.
ac+ad+bc+bd

 

[Pere Callahan]

Why would you suspect this not to be true?

If you're in doubt re-check the definition of a ring.

 

[apalmer3]

I doubt it simply because I'm in a 4000 level Math course. It can't be this easy an answer.

 

[Dick]

What else could it be?

 

[Pere Callahan]

Well, if there's no more information about a,b,c,d than I don't know what else you could do

 

[apalmer3]

Okay, so I'm going to guess that everyone agrees with me that this is right? It just seems too easy! Oh well, I'll accept it and move on. :-D

 

[PingPong]

Treat (a+d) as one element and use the distributive property of rings. Then use it again. Make sure you keep the ordering if you're not dealing with a commutative ring.

 

[apalmer3]

Okay, that makes sense, PingPong! So, technically, the order is different.

(a+b)(c+d)
=> (a+b)c+(a+b)d
=> ac+bc+ad+bd

 

[PingPong]

Right, but remember that in a ring, addition is commutative, so your order doesn't matter. What I said about ordering was meant to be applied to multiplication (unless you're dealing with a commutative ring). So what you had at the beginning was correct, but now you see why!

EDIT: Actually as a side note, commutativity of addition in a ring is an unneeded axiom, because it follows from the distributive property. If you treat (c+d) as one element, the distributivity property gives (a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd. This means that ac+bc+ad+bd (your method) is equivalent to ac+ad+bc+bd, or ad+bc=bc+ad. Thus addition is commutative.