Generating set of a module: Difference between revisions

Revision as of 09:15, 12 January 2022

Revision as of 11:09, 26 January 2022

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Put in another way, there is a [[surjection]]

: <math>\bigoplus_{g \in G\Gamma} R \to M, \, r_g \mapsto r_g g,</math>

where we wrote ''r''_''g'' for an element in the ''g''-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g. ''M'' itself, this shows that a module is a [[quotient module|quotient]] of a [[free module]], a useful fact.)