Editing Semisimple module (section)

== Semisimple rings ==

A ring is said to be (left-)'''semisimple''' if it is semisimple as a left module over itself.{{sfn|ps=|Sengupta|2012|p=125}} Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity.

A semisimple ring may be characterized in terms of [[homological algebra]]: namely, a ring ''R'' is semisimple if and only if any [[short exact sequence]] of left (or right) ''R''-modules splits. That is, for a short exact sequence
: <math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 </math>
there exists {{nowrap|''s'' : ''C'' → ''B''}} such that the composition {{nowrap|''g'' ∘ ''s'' : ''C'' → ''C''}} is the identity. The map ''s'' is known as a section. From this it follows that
: <math>B \cong A \oplus C</math>
or in more exact terms
: <math>B \cong f(A) \oplus s(C).</math>

In particular, any module over a semisimple ring is [[injective module|injective]] and [[projective module|projective]]. Since "projective" implies "flat", a semisimple ring is a [[von Neumann regular ring]].

Semisimple rings are of particular interest to algebraists. For example, if the base ring ''R'' is semisimple, then all ''R''-modules would automatically be semisimple. Furthermore, every simple (left) ''R''-module is isomorphic to a minimal left ideal of ''R'', that is, ''R'' is a left [[Kasch ring]].

Semisimple rings are both [[Artinian ring|Artinian]] and [[Noetherian ring|Noetherian]]. From the above properties, a ring is semisimple if and only if it is Artinian and its [[Jacobson radical]] is zero.

If an Artinian semisimple ring contains a field as a [[Center of a ring|central]] [[subring]], it is called a [[semisimple algebra]].

=== Examples ===
* For a [[commutative ring]], the four following properties are equivalent: being a [[semisimple ring]]; being [[Artinian ring|artinian]] and [[reduced ring|reduced]];{{sfn|ps=|Bourbaki|2012|p=133|loc=VIII}} being a [[reduced ring|reduced]] [[Noetherian ring]] of [[Krull dimension]] 0; and being isomorphic to a finite direct product of fields.
* If ''K'' is a field and ''G'' is a finite group of order ''n'', then the [[group ring]] ''K''[''G''] is semisimple if and only if the [[characteristic (algebra)|characteristic]] of ''K'' does not divide ''n''. This is [[Maschke's theorem]], an important result in [[group representation theory]].
* By the [[Wedderburn–Artin theorem]], a unital ring ''R'' is semisimple if and only if it is (isomorphic to) {{nowrap|M<sub>''n''<sub>1</sub></sub>(''D''<sub>1</sub>) × M<sub>''n''<sub>2</sub></sub>(''D''<sub>2</sub>) × ... × M<sub>''n''<sub>''r''</sub></sub>(''D''<sub>''r''</sub>)}}, where each ''D''<sub>''i''</sub> is a [[division ring]] and each ''n''<sub>''i''</sub> is a positive integer, and M<sub>''n''</sub>(''D'') denotes the ring of ''n''-by-''n'' matrices with entries in ''D''.
* An example of a semisimple non-unital ring is M<sub>∞</sub>(''K''), the row-finite, column-finite, infinite matrices over a field ''K''.

=== Simple rings ===
{{main|Simple ring}}

One should beware that despite the terminology, ''not all simple rings are semisimple''. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if ''R'' is a simple ring with a minimal left/right ideal, then ''R'' is semisimple.

Classic examples of simple, but not semisimple, rings are the [[Weyl algebra]]s, such as the '''Q'''-algebra
: <math> A=\mathbf{Q}{\langle x,y \rangle }/\langle xy-yx-1\rangle\ ,</math>
which is a simple noncommutative [[domain (ring theory)|domain]].  These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings.  The [[module theory]] for the Weyl algebras is well studied and differs significantly from that of semisimple rings.

=== Jacobson semisimple ===
{{main|Semiprimitive ring}}
A ring is called ''Jacobson semisimple'' (or ''J-semisimple'' or ''[[semiprimitive ring|semiprimitive]]'') if the intersection of the maximal left ideals is zero, that is, if the [[Jacobson radical]] is zero.  Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.  A J-semisimple ring is semisimple if and only if it is an [[artinian ring]], so semisimple rings are often called ''artinian semisimple rings'' to avoid confusion.

For example, the ring of integers, '''Z''', is J-semisimple, but not artinian semisimple.