Jacobson ring [SEP] named after
Jacobson ring
In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals.
Jacobson rings were introduced independently by , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by , who named them Hilbert rings after David Hilbert because of their relation to Hilbert's
Zorn ring
In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent "x" there exists an element "y" such that "xy" is a non-zero idempotent . named them after Max August Zorn, who studied a similar condition in .
For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J("R") is a nil ideal and every right ideal of "R" which is not contained in J("R") contains a