Generating set of a module: Difference between revisions

Revision as of 14:04, 17 August 2019

Revision as of 00:46, 26 April 2020

Line 11:

where we wrote ''r''_''g'' for an element in the ''g''-th component of the direct sum. (Coincidentally, since a generating set always exists; for example, ''M'' itself, this shows that a module is a 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 ita 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>

The cardinality of a minimal generating set need not be an invariant of the module; '''Z''' is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set {{nowrap|{ 2, 3 }}}. What is uniquely determined by a module is the [[infimum]] of the numbers of the generators of the module.