- #1
chaotixmonjuish
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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
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