Exercises in Classical Ring Theory pp 35-68 | Cite as

# Jacobson Radical Theory

Chapter

## Abstract

The Jacobson radical of a ring *R*, denoted by rad *R*, is the intersection of the maximal left ideals of *R*. This notion is left-right symmetric; in particular, rad *R* is an ideal of *R*. A good way to understand rad *R* is to think of it as the ideal of elements annihilating all left (resp. right) simple *R*-modules. The Jacobson radical is also closely tied in with *U*(*R*), the group of units of *R*. In fact, rad *R* is the largest ideal *R* such that 1 + *R* ⊆ *U*(*R*).

## Keywords

Prime Ideal Maximal Ideal Direct Summand Left Ideal Group Ring## Preview

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## Copyright information

© Springer Science+Business Media New York 1995