# Semiprimitive ring

From Commalg

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 ringsVIEW 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

## Contents

## 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