Abstract
We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings.
Highlights
Given a ring R, an ideal P of R is called left primitive if there is a simple left R-module M such that P is the ideal of elements r ∈ R that annihilate M ; i.e., P = {r ∈ M : rm = 0 ∀m ∈ M }
We note that the intersection of the primitive ideals of a ring is equal to the Jacobson radical—this is true whether one works with left or right primitive ideals [24, Prop. 3.16]
One of the most appealing results in terms of characterizing primitive ideals in rings is the work of Dixmier and Moeglin [21, 47], who showed that if L is a finite-dimensional complex Lie algebra the primitive ideals of the enveloping algebra U (L) are just the prime ideals of Spec(U (L)) that are locally closed in the Zariski topology
Summary
Given a ring R, an ideal P of R is called left primitive if there is a simple left R-module M such that P is the ideal of elements r ∈ R that annihilate M ; i.e.,. The notion of a right primitive ideal can be defined analogously. For the remainder of this paper, we will speak only of primitive ideals, with the understanding that we are always working on the left, and, in any case, for most of the rings considered in this survey, a left/right symmetric characterization of primitivity is given. We note that the intersection of the primitive ideals of a ring is equal to the Jacobson radical—this is true whether one works with left or right primitive ideals [24, Prop. We note that the intersection of the primitive ideals of a ring is equal to the Jacobson radical—this is true whether one works with left or right primitive ideals [24, Prop. 3.16]
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