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&mdash;the analogue of a [[cyclic group]] for modules.
 
element&mdash;the analogue of a [[cyclic group]] for modules.
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In a left <math>R</math>-module <math>M</math>, the cyclic [[submodule]] generated by an element
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<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 $R$-module over a ring $R$) is a module that is generated by a single element—the analogue of a cyclic group for modules.

In a left $R$-module $M$, the cyclic submodule generated by an element $\alpha$ is often denoted $\langle \alpha \rangle$.

Every cyclic left $R$-module is isomorphic to a quotient module of the left-regular module over $R$ (that is, a quotient module of $R$ as a left $R$-module).

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See also