Editing Jacobson radical (section)

== Definitions ==
There are multiple equivalent definitions and characterizations of the Jacobson radical, but it is useful to consider the definitions based on if the ring is [[commutative ring|commutative]] or not.

=== Commutative case ===
In the commutative case, the Jacobson radical of a commutative ring ''R'' is defined as<ref>{{Cite web|title=Section 10.18 (0AMD): The Jacobson radical of a ring—The Stacks project|url=https://stacks.math.columbia.edu/tag/0AMD|access-date=2020-12-24|website=stacks.math.columbia.edu}}</ref> the [[intersection (set theory)|intersection]] of all [[maximal ideal]]s <math>\mathfrak{m}</math>. If we denote {{nowrap|Specm ''R''}} as the set of all maximal ideals in ''R'' then<blockquote><math>\mathrm{J}(R) = \bigcap_{
    \mathfrak{m} \,\in\, \operatorname{Specm}R
} \mathfrak{m}</math></blockquote>This definition can be used for explicit calculations in a number of simple cases, such as for [[local ring]]s {{nowrap|(''R'', <math>\mathfrak{p}</math>)}}, which have a unique maximal ideal, [[Artinian ring]]s, and [[product ring|products]] thereof. See the examples section for explicit computations.

=== Noncommutative/general case ===
For a general ring with unity ''R'', the Jacobson radical J(''R'') is defined as the ideal of all elements {{nowrap|''r'' ∈ ''R''}} such that {{nowrap|1=''rM'' = 0}} whenever ''M'' is a [[simple module|simple]] ''R''-module. That is,
<math display="block">\mathrm{J}(R) = \{r \in R \mid rM = 0 \text{ for all } M \text{ simple} \}.</math>
This is equivalent to the definition in the commutative case for a commutative ring ''R'' because the simple modules over a commutative ring are of the form {{nowrap|''R'' / <math>\mathfrak{m}</math>}} for some maximal ideal {{nowrap|<math>\mathfrak{m}</math> of ''R''}}, and the [[annihilator (ring theory)|annihilators]] of {{nowrap|''R'' / <math>\mathfrak{m}</math>}} in ''R'' are precisely the elements of <math>\mathfrak{m}</math>, i.e. {{nowrap|1=Ann<sub>''R''</sub>(''R'' / <math>\mathfrak{m}</math>) = <math>\mathfrak{m}</math>}}.