Semiprimitive ring: Difference between revisions
(New page: {{curing property}} ==Name== A '''semiprimitive ring''' or '''ring with trivial Jacobson radical''' is sometimes termed a '''semisimple ring''', although the latter term is usually reser...) |
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Revision as of 14:13, 6 May 2008
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
Name
A semiprimitive ring or ring with trivial Jacobson radical is sometimes termed a semisimple ring, although the latter term is usually reserved for a semisimple Artinian ring.
Definition
Symbol-free definition
A commutative unital ring is said to be semiprimitive or to have trivial Jacobson radical if it satisfies the following equivalent conditions:
- Its Jacobson radical (i.e., the intersection of all its maximal ideals) is the zero ideal
- It is a subdirect product of fields i.e., it can be embedded inside a direct product of fields