Sum of Ideals of the Same Type

Question

I have two questions:

1) Is a finite sum of idempotent ideals of a ring $R$ idempotent?

2) Is any sum of nil ideals of a ring $R$ nil?

As far as I know, a finite sum of nil ideals of a commutative ring $R$ is nil too by Koethe conjecture which is true for such rings.

user26857 · Accepted Answer · 2016-03-20 01:05:08Z

2

2) Every element which belongs to a sum of ideals belongs to a finite sum of them.

answered Mar 19, 2016 at 23:39

user26857

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karparvar · Accepted Answer · 2016-03-20 00:51:54Z

1

1) If $I$ and $J$ are idempotent ideals of $R$ then we have:

$(I+J)^2=I^2+IJ+JI+J^2=I+J$, since $IJ⊆I=I^2$ and also, $JI⊆J=J^2$. The general result would follow by induction.

answered Mar 20, 2016 at 0:51

karparvar

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$\begingroup$ @user26857 I think that one could well generalize my solution to an arbitrary sum of idempotent ideals. In fact, if $S=\sum _t I_t$ where $t$ runs in an arbitrary index set so that $I_t$'s are idempotent ideals then $S^2=\sum_t {I_t}^2+\sigma$ where $\sigma$ is a sum each of whose summands is an ideal contained in some $I_t$. Hence, $S^2=\sum_t I_t+\sigma=S$. $\endgroup$
– karparvar
Mar 20, 2016 at 17:53

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2 Answers 2