Algebra Proofs

Algebra Proofs!!

I have two questions just to help verify what I'm doing:

Let R be a commutative ring with identity. Prove that R is an integral domain if and only if cancellation holds in R (that is, a no equal to 0 and ab=ac in R imply b=c)

=> Suppose cancellation holds: ab=0 -> ab=0a -> a isn't 0 so b=0

<= Since a doesn't equal 0
ab=ac -> ab-ac=0 -> a(b-c)=0 -> b-c=0 -> b=c cancellation holds

alternatively, could the second part be proven as such:

ab=ac
a^-1ab=aca^-1
b=c
cancellation holds

My second question is: How do I go about proving the following

(a^m)(a^n)=a^m+n and (a^m)^n=a^mn
 

HallsofIvy

Science Advisor
41,626
821
Re: Algebra Proofs!!

Your "alternative" cannot be used because a-1 does not necessarily exist!

For your second question, I would suggest fixing m and using induction on n.
 

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top