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primitive ideal (Definition)

Let $ R$ be a ring, and let $ I$ be an ideal of $ R$. We say that $ I$ is a left (right) primitive ideal if there exists a simple left (right) $ R$-module $ X$ such that $ I$ is the annihilator of $ X$ in $ R$.

We say that $ R$ is a left (right) primitive ring if the zero ideal is a left (right) primitive ideal of $ R$.

Note that $ I$ is a left (right) primitive ideal if and only if $ R/I$ is a left (right) primitive ring.



"primitive ideal" is owned by antizeus.
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Other names:  primitive ring
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Cross-references: zero ideal, annihilator, simple, right, ideal, ring
There are 5 references to this entry.

This is version 2 of primitive ideal, born on 2001-11-24, modified 2002-10-25.
Object id is 1007, canonical name is PrimitiveIdeal.
Accessed 4005 times total.

Classification:
AMS MSC: 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
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