Module theory MOC

Hom-module

Let 𝑅 be a commutative ring and π‘ˆ,𝑉 be 𝑅-modules. Then the set π‘…π–¬π—ˆπ–½(π‘ˆ,𝑉) of 𝑅-module homomorphisms is itself equipped with the structure of an 𝑅-module, i.e. we have an internal hom-functor

π‘…π–¬π—ˆπ–½(βˆ’,βˆ’):π‘…π–¬π—ˆπ–½π¨π©Γ—π‘…π–¬π—ˆπ–½β†’π‘…π–¬π—ˆπ–½

which is the right adjoint of the tensor product, i.e.

(βˆ’)βŠ—π‘…π‘‰βŠ£π‘…π–¬π—ˆπ–½(𝑉,βˆ’)


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