Difference between revisions of "Simple module"
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| − | A '''simple module''' over a [[ring]] <math>R</math> is a [[module]] that is simple as a [[group with operators]]—that is, it is a module with no [[submodule]]s other than itself and the zero module, and it is not itself the zero module. | + | A '''simple module''' over a [[ring]] <math>R</math> is a [[module]] that is simple as a [[group with operators]]—that is, it is a module with no [[submodule]]s other than itself and the zero module, and it is not itself the zero module and scalar products are not all equal to 0. |
If <math>R</math> is a [[commutative ring]], then every simple module over <math>R</math> is | If <math>R</math> is a [[commutative ring]], then every simple module over <math>R</math> is | ||
Latest revision as of 10:02, 29 September 2012
A simple module over a ring 
is a module that is simple as a group with operators—that is, it is a module with no submodules other than itself and the zero module, and it is not itself the zero module and scalar products are not all equal to 0.
If is a commutative ring, then every simple module over is
isomorphic (as an -module) to a quotient ring of by a
maximal ideal; that is,
every simple module over is isomorphic (as an -module) to a quotient
ring of that is a field. This is not the case when is not
commutative. In this case, every simple left -module is isomorphic
(as a left -module) to the quotient of by a maximal left ideal.
For example, all simple modules over the ring of integers 
are of the form 
, where 
is a prime. A more
interesting example of a simple module is the (left) module of
complex numbers over the ring 
of
complex numbers with a noncommuting indeterminate 
adjoined, where
corresponds to complex conjugation.
