Tensor product of modules: Difference between revisions

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In [[mathematics]], the '''tensor product of modules''' is a construction that allows arguments about [[bilinear map|bilinear]] maps (e.g. multiplication) to be carried out in terms of [[module homomorphism|linear map]]s. The module construction is analogous to the construction of the [[tensor product]] of [[vector space]]s, but can be carried out for a pair of [[module (mathematics)|modules]] over a [[commutative ring]] resulting in a third module, and also for a pair of a right-module and a left-module over any [[ring (mathematics)|ring]], with result an [[abelian group]]. Tensor products are important in areas of [[abstract algebra]], [[homological algebra]], [[algebraic topology]], [[algebraic geometry]], [[operator algebras]] and [[noncommutative geometry]]. The [[universal property]] of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via [[linear operator|linear operations]]. The tensor product of an algebra and a module can be used for [[extension of scalars]]. For a commutative ring, the tensor product of modules can be iterated to form the [[tensor algebra]] of a module, allowing one to define multiplication in the module in a universal way.

==Balanced product==