For a commutative ring with identity, we know that the annihilator (as a function on ideals) of the sum of some ideals equals to the intersection of their annihilators, I am looking for a function on ideals, say $F$, of a commutative ring that $F$ of the intersection of some ideals equals to the sum $F$ of them.
2026-03-25 11:52:57.1774439577
A function on ideals that converts intersection to sum
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The best result I know in this direction (still using the annihilator map) is in the work of G. F. Birkenmeier, M. Ghirati & A. Taherifar.
For instance, this determined that among semiprime rings, the quasi-Baer ones had the property you’re describing.
I haven’t really seen any other order reversing maps with the property you’re speaking of.