Generating set of a module: Difference between revisions

Revision as of 11:09, 26 January 2022

Revision as of 01:01, 29 May 2022

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: <math>x = r_1 g_1 + \cdots + r_m g_m.</math>

Put in another way, there is aan ''R''-linear [[surjection]]

: <math>\bigoplus_{g \in \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.)

A generating set of a module is said to be '''minimal''' if no [[proper subset]] of the set generates the module. If ''R'' is a [[field (mathematics)|field]], then a minimal generating set is the same thing as a [[basis (linear algebra)|basis]]. Unless the module is [[finitely-generated module|finitely-generated]], there may exist no minimal generating set.<ref>{{cite web|url=https://mathoverflow.net/q/33540 |title=ac.commutative algebra – Existence of a minimal generating set of a module – MathOverflow|work=mathoverflow.net}}</ref>