Math 2.0

Commutative algebra Definition

Finitely generated module

Builds on Module over a ring

Definition

Let RR be a ring with identity element and let MM be a left module over RR. The module MM is finitely generated when there are x1,…,xn∈Mx_1, \dots, x_n \in M such that each x∈Mx \in M is x=r1x1+⋯+rnxnfor some r1,…,rn∈R.x = r_1 x_1 + \cdots + r_n x_n \qquad \text{for some } r_1, \dots, r_n \in R.

Remarks

Over a field, the finitely generated modules are the finite-dimensional vector spaces. ℤn\Z^n and ℤ/nℤ\Z / n\Z are finitely generated over ℤ\Z; the group ℚ\Q is not. Over a Noetherian ring, each submodule of a finitely generated module is finitely generated. The finitely generated modules over a principal ideal domain are classified by the structure theorem.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Finitely_generated_module, Definition
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MSC13E15, 16D10
DefinesFinitely generated module
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