Re: JSH: The "Published" paper he dosen't what you to know about.

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[William Hughes]
As far as ideals I and J go the standard defintion seems to be

i. I and J are coprime if there is no prime ideal
containing both I and J.

An alternate definition, equivalent for rings with multiplicative
identityis

ii. I and J are coprime if I+J=R

but i and ii are not (may not be?) the same for rings without
multiplicative identity.

[Arturo Magidin]
If I+J = R, then I and J are necessarily coprime by definition
i. However, it is possible for I and J to be coprime by definition i
and not by definition ii in the case of rings without identity. The
example in R=2Z are the ideals (2)=4Z and (4)=8Z. No prime ideal
contains both 4Z and 8Z, but I+J = 4Z which is not equal to R.

More, in R=2Z no two /elements/ are coprime by (the inappropriate) ii: pick
any two elements i and j of R, and any two multipliers i' and j' from R.
These are all even, so i*i' + j*j' is divisible by 4, so in particular 2
isn't a possible sum (or -2, or 6, or ...). That is, there are no elements
i and j such that iR + jR = R. There are nevertheless coprime ideals in the
sense of ii (like 4Z + 6Z = 2Z = R; 4Z = 2R, but 6Z =/= iR for any i in R).

[... more excellent reasons to avoid rings without identity ;-) ...]


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