- #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?
Last edited by a moderator: