2
$\begingroup$

I am looking for some references that contained a study of ideals with the following *-property:

Let $I $ be an ideal of a commutative ring with ideantity. The ideal $I $ has the *-property if $I\subseteq \sum_{ i\in A }I_i $, where $\{ I_i \}_{ i\in A } $ is a family of ideals of $R $, then there exists $j\in A $ such that $I\subseteq I_j $.

Improve this question
$\endgroup$
9
  • 1
    $\begingroup$ Can you give an example other than the zero ideal? $\endgroup$
    – user1688
    Commented Jan 30, 2018 at 13:12
  • 2
    $\begingroup$ In a local ring if the maximal ideal is pricipal it has the *- property. $\endgroup$
    – user119996
    Commented Jan 30, 2018 at 14:07
  • 1
    $\begingroup$ @Corbennick Well, one obvious example would be to take an Artinian ring whose ideals are linearly ordered. The unique minimal ideal $I$ would satisfy this condition. $\endgroup$
    – rschwieb
    Commented Jan 30, 2018 at 14:41
  • $\begingroup$ I think I recall considering the "finite" version of this question before ($|A|$ being finite) in dual rings. I think i was looking for ways to dualize the definition of "prime ideal," and the annihilator map was the candidate. I don't recall finding special names for it. $\endgroup$
    – rschwieb
    Commented Jan 30, 2018 at 14:48
  • 1
    $\begingroup$ @ZachTeitler Clearly for Noetherian uniserial rings, sure. I don't think your claim is true though in non-Noetherian ones, though. It seems like if you take an infinite strictly ascending chain, and the union $I$ would be contained in the sum (=union) of the chain, and yet it wouldn't be contained in any of the elements of the chain. $\endgroup$
    – rschwieb
    Commented Jan 30, 2018 at 16:30

0

Reset to default

You must log in to answer this question.