Difference between revisions of "Cyclic module"
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over a [[ring]] <math>R</math>) is a [[module]] that is generated by a single | over a [[ring]] <math>R</math>) is a [[module]] that is generated by a single | ||
element—the analogue of a [[cyclic group]] for modules. | element—the analogue of a [[cyclic group]] for modules. | ||
| + | |||
| + | In a left <math>R</math>-module <math>M</math>, the cyclic [[submodule]] generated by an element | ||
| + | <math>\alpha</math> is often denoted <math>\langle \alpha \rangle</math>. | ||
Every cyclic left <math>R</math>-module is [[isomorphic]] to a quotient module of the | Every cyclic left <math>R</math>-module is [[isomorphic]] to a quotient module of the | ||
Latest revision as of 15:09, 17 August 2009
A cyclic module (or more specifically, a cyclic left 
-module
over a ring ) is a module that is generated by a single
element—the analogue of a cyclic group for modules.
In a left -module 
, the cyclic submodule generated by an element
is often denoted 
.
Every cyclic left -module is isomorphic to a quotient module of the
left-regular module over (that is, a quotient module of
as a left -module).
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