- #1

Silversonic

- 130

- 1

I can't wrap my head around this proof that the sum of two nilpotent ideals is nilpotent, I get stuck at one stage:

http://imageshack.com/a/img706/5732/5wgq.png

I'm fine with every except showing by induction [itex] (I+J)^{N+k} = I^k \cap J + I \cap J^k [/itex]. Here's my attempt;

Base case: k = 1,

[itex] (I+J)^{N+1} = [I+J, (I+J)^N] \subseteq [I+J,I \cap J] = [I, I \cap J] + [J, I \cap J] \subseteq I \cap J + I \cap J [/itex]

since [itex] [I, I \cap J], [J, I \cap J] \subseteq I \cap J [/itex] as [itex] I \cap J [/itex] is an ideal.

Now inductive step;

[itex] (I+J)^{N+k+1} = [I+J, (I+J)^{N+k}] = [I+J, I^k \cap J + I \cap J^k] = [I, I^k \cap J] + [J, I^k \cap J] + [I,I \cap J^k] + [J,I \cap J^k] [/itex]

Now it's easy to see

[itex] [I, I^k \cap J] \subseteq I^{k+1} \cap J [/itex]

[itex] [J, I \cap J^k] \subseteq I \cap J^{k+1} [/itex]

But I have no idea what I can do with the [itex] [J, I^k \cap J] + [I,I \cap J^k] [/itex] term so that it reduces to the form I want. Any help?

http://imageshack.com/a/img706/5732/5wgq.png

I'm fine with every except showing by induction [itex] (I+J)^{N+k} = I^k \cap J + I \cap J^k [/itex]. Here's my attempt;

Base case: k = 1,

[itex] (I+J)^{N+1} = [I+J, (I+J)^N] \subseteq [I+J,I \cap J] = [I, I \cap J] + [J, I \cap J] \subseteq I \cap J + I \cap J [/itex]

since [itex] [I, I \cap J], [J, I \cap J] \subseteq I \cap J [/itex] as [itex] I \cap J [/itex] is an ideal.

Now inductive step;

[itex] (I+J)^{N+k+1} = [I+J, (I+J)^{N+k}] = [I+J, I^k \cap J + I \cap J^k] = [I, I^k \cap J] + [J, I^k \cap J] + [I,I \cap J^k] + [J,I \cap J^k] [/itex]

Now it's easy to see

[itex] [I, I^k \cap J] \subseteq I^{k+1} \cap J [/itex]

[itex] [J, I \cap J^k] \subseteq I \cap J^{k+1} [/itex]

But I have no idea what I can do with the [itex] [J, I^k \cap J] + [I,I \cap J^k] [/itex] term so that it reduces to the form I want. Any help?

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