Simple Rings: Commutativity and Identity

[jem05]

Hello everyone,

i was checking out a paper on simple rings
http://www.imsc.res.in/~knr/RT09/sssrings.pdf
and they said that all commutative simple rings are fields.
i just don't see why they should have identity.
thank you.

 

[arkajad]

Let R be a simple ring. Let x be a nonzero element of R. Then Rx is a nonzero ideal Thus Rx=R. Thus there exists y in R such that yx=1. Thus x is invertible.

 

[jem05]

but i don't have identity in R.
It's not given, i think we need to show it, right?

 

[arkajad]

From Wikipedia's definition of a ring:

3. Existence of multiplicative identity. There exists an element 1 in R, such that for all elements a in R, the equation 1 · a = a · 1 = a holds.

 

[jem05]

yeah i agree with this,
but sorry i don't see how this is equivalent to what you did.
what is our 1 here?

after i get that R has identity,
then since {0} is a maximal ideal then R/ {0} = R is a field and I am done

 

[arkajad]

Every ring, by definition, has a multiplicative identity that is usually denoted as 1. So where is your problem? We have proven that each simple commutative ring is a field using only the definitions and nothing more. If you still have some problem with this proof - please, be very precise. For instance I fail to understand your "{0} being a maximal ideal".

 

[Landau]

Every ring, by definition, has a multiplicative identity that is usually denoted as 1. So where is your problem?

The problem might be that not everyone requires a ring to have an identity; it seems that jem05 is one of them. It also seems that the paper is NOT one of them (as most texts on modules and commutative algebra), so I don't see a problem any more.

\\edit: I see that jem05 still has problems with the proof after assuming R has an identity. Then I agree: please be precise about what you don't understand.

 

[arkajad]

The problem might be that not everyone requires a ring to have an identity; it seems that jem05 is one of them. It also seems that the paper is NOT one of them (as most texts on modules and commutative algebra), so I don't see a problem any more.

Yeah, good point. In fact, this should have been the first thing that needed to be made clear.

 

[arkajad]

For commutative rings without identity http://books.google.fr/books?id=H0m... commutative ring "without identity"&f=false" observations may be relevant.

 

[jem05]

ok, thanks
yeah sorry for not being clearer, but yeah i assumed i have no identity for my ring R