Cyclic module

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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