Help with abstract algebra proof

[jus8727]

prove that a finite ring with identity has characteristic n for some n>0.
been trying for a while getting nowhere any ideas?

 

[Dick]

What have you been trying? The ring is FINITE. If f(n)=1+1+1+... n times, you can't get different values of f(n) for all n.

 

[jus8727]

you are assuming that we are dealing with numbers and with our normal addition. addition can be defined in any way. that's my problem just because it is a ring does not mean we are dealing with numbers or at the least the regular def of addition and multiplication

 

[Dick]

Your ring has an identity. I called that '1'. Your ring also has an addition operation, I called that '+'. Why do you think I'm assuming anything?

 

[jus8727]

how do u know what happens when u add to numbers, how do u know that they get larger define larger. we may not be dealing with numbers all we know is that its a ring. we don't know what 1 is it is just the muliplictive inverse. it may not be a number

 

[Dick]

Who said that anything was larger? I just said the f(n) couldn't ALL be different because your ring is finite. Besides 1 is a multiplicative identity, not an inverse. I really don't think you are in a mood to listen to any advice here.

 

[jus8727]

im in a good mood sorry if u get affended i really appericate the help but i dissagree i a few things

 

[jus8727]

how to u do know that 1+1+1...n times isn't just one? we don't know how addition is defined?

 

[Dick]

Ok, sorry to be thin skinned, but you objecting to things I'm not even saying while not paying any attention to my original suggestion. Once more, your ring R is FINITE. That means f(n)=f(m) for some n and m. See post 2 for the definition of f(n). What does that tell you about characteristic? If you want to object to more things that's ok, because I'm not here. I'm going to bed.

 

[Dick]

how to u do know that 1+1+1...n times isn't just one? we don't know how addition is defined?

If 1+1+... n times=1, then 1+1+.. n-1 times=0. We do know that much about how addition is defined. Characteristic!

 

[AlexChandler]

Since the ring is finite, we know that there exist some integers n, and m, such that n*1 is equal to m*1, and let n>m. (this was already suggested) Try taking the difference of these to find if there is a characteristic.