Re: an ideal problem

In article <Pine.BSI.4.58.0505250337200.25022@xxxxxxxxxxxxxxxxx>,
William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:
>On Wed, 25 May 2005, Per Vognsen wrote:
>
>> Li Yi wrote:
>> > I and J are ideals of a ring R such that I + J = R.
>> >
>> > (A ring here is assumed to have commutativity and multiplicative
>> > identity)
>> >
>> > Show that I /\ J = IJ.
>>
>> The inclusion IJ sub I cap J is trivial and does not require the
>> hypothesis.
>>
>If x in IJ, some a in I, b in J with x = ab
> ab in I, ab in J, ab in I /\ J

This is incorrect. The best you can say is that

x = i_1j_1 + ... + i_kj_k

for some nonnegative integer k, i_r in I, j_r in J.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

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