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/sci/ - Science & Math

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4404223 No.4404223 [Reply] [Original]

Let I1 ⊂ I2 ⊂ I3 ⊂ . . . be ideals in a ring R. Show that I, the union from n = 1 to infinity of In, is also
an ideal.

I real confused /sci/.

For a start, when it says the union from 1 to infinity... do I just show that for every given n, the union is an ideal?

Also... if that's the case, this question is retarded, because clearly for any n the union is just In, which is trivially an ideal.

>> No.4404243

>do I just show that for every given n, the union is an ideal?
No

This would be like saying
0.999...<1
because for any n
<div class="math">0.\underbrace{99...9}_n <1</div>

>> No.4404269

Okay well, that's a sum, I'm talking about unions.

Your comment is fairly useless as you didn't actually say what was incorrect about my definition of an infinite union and how to define it correctly.

>> No.4404282

Applying zorn's lemma huh?
Anyway, you just need to show that for any element x of the union, rx is in the union right?
So you say:
x is in the union, so x is in I_n for some n, so rx is also in I_n, so rx is in the union.
I guess you also have to show its a subgroup of R or some shit, in which case you have two elements x and y and you have to pick the higher numbered ideal so that it contains both x and y, but its the same idea.

>> No.4404284

>>4404243
I don't see what's wrong with that logic.

>> No.4404286

i think it asks you to do use induction

>> No.4404295

Just think of the definition of an arbitrary union (or a countable union in this case). To say r is in U (I_n) from 1 to inf means there exists some n in the natural numbers such that r is in I_n. So you need to to show the union is an additive subgroup of R, also that r an element of union implies r*x is an element of union for all x in R.

>> No.4404299

>>4404284
Limit points of a set do not have to be contained within the set itself, and therefore do not necessarily have the same properties.

>> No.4404309

>>4404295
> that feel when second year cambridge maths and they haven't taught us this yet.

>> No.4404401

Okay, thanks bros. It was still retardedly easy... in fact I wasn't really that far off to start with.

Gonna try the second part of the question now, don't expect it to be much harder though.

>> No.4404414

>>4404295
You the guy who posted in the group theory question thread too?
You seem to really like right actions.