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Natural isomorphism between Hom out of a tensor product and Hom into a Hom-module.
Tensor–Hom adjunction lemma: Let be unital rings, let be an -bimodule, let be a left -module, and let be a left -module. Then there is a natural isomorphism of abelian groups
functorial in and .
This is the concrete form of the Tensor–Hom adjunction for a bimodule, relating tensor products and Hom-modules.
Let be rings. An -bimodule is an abelian group that is simultaneously a left module over and a right module over , such that the actions are compatible: for all , , and , one has .
Bimodules are the natural setting for many constructions (e.g. balanced products) and are the input for the tensor product, where compatibility of left/right actions is essential.
Let be a ring (often assumed a unital ring). A left -module is an abelian group together with a scalar multiplication map , , such that for all and :
A right -module is defined similarly with a map , , satisfying the analogous axioms.
The axioms are collected in module axioms. When is a field, left -modules are the same objects as vector spaces. Ideals of a ring give basic examples of modules, linking module theory to ideal theory.
The tensor–Hom adjunction (over a commutative ring ) is the natural isomorphism of abelian groups (indeed -modules)
natural in , where is the tensor product and is the Hom module.
The tensor product of modules of a right -module and a left -module (for a ring ) is an abelian group equipped with a canonical balanced bilinear map , , satisfying the universal property: every balanced bilinear map out of factors uniquely through a homomorphism out of .
This construction is functorial in both variables and is central for “extension of scalars” and for measuring non-exactness via derived functors. Over commutative rings it specializes to the tensor product of left modules.
A Hom module is the set of all module homomorphisms between -modules, equipped with pointwise addition . This makes an abelian group. If is a commutative ring, then is naturally an -module via .
In the noncommutative setting, additional module structures arise from bimodules: if is an -bimodule and is a left -module, then carries a natural left -module structure by .