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单词 module
释义

module

An equivalent idea to a vector space but where scalars are from a ring which need not be a field. Given a ring R with a multiplicative identity 1, an R-module M is an abelian group together with a notion of scalar multiplication (r,m) ↦r∙m, which further satisfies

for all r1,r2,r in R and m1,m2,m in M. Typically R is further assumed to be a principal ideal domain or a Euclidean domain. ℤ-modules are abelian groups and vice versa. An F[x]-module, where F is a field, is a vector space V over F together with a linear map x:V → V defining scalar multiplication by x. Modules are intimately connected with representation theory. See free module, Jordan normal form, rational canonical form, Smith normal form, structure theorem, torsion element.

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更新时间:2025/4/30 4:15:32