Content deleted Content added
more concise and clearer Tag: Reverted |
Undid that: I need to make it clear without losing the sense of what operation is performed |
||
Line 7:
: <math>x = r_1 g_1 + \cdots + r_m g_m.</math>
Put in another way, there is
: <math>\bigoplus_{g \in \Gamma} R \to M
where we wrote ''r''<sub>''g''</sub> 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>
| |||