Let M be a left R-module. (i) Prove that the map φM: HomR(R, M) →M, given by φM: f ↦f(1), is an

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Let M be a left R-module. (i) Prove that the map φM: HomR(R,...

Let $M$ be a left $R$-module. (i) Prove that the map $\varphi_{M}: \operatorname{Hom}_{R}(R, M) \rightarrow M$, given by $\varphi_{M}: f \mapsto f(1)$, is an $R$-isomorphism. Hint. Make the abelian group $\operatorname{Hom}_{R}(R, M)$ into a left $R$ module by defining $r f$ (for $f: R \rightarrow M$ and $r \in R$ ) by $r f: s \mapsto f(s r)$ for all $s \in R .$ (ii) If $g: M \rightarrow N$, prove that the following diagram commutes: Conclude that $\varphi=\left(\varphi_{M}\right)_{M \in o b j(R M o d)}$ is a natural isomorphism from $\operatorname{Hom}_{R}(R, \square)$ to the identity functor on ${ }_{R} \mathbf{M o d}$. [Compare with Example 1.16(ii).]

Let $M$ be a left $R$-module.
(i) Prove that the map $\varphi_{M}: \operatorname{Hom}_{R}(R, M) \rightarrow M$, given by $\varphi_{M}: f \mapsto f(1)$, is an $R$-isomorphism.
Hint. Make the abelian group $\operatorname{Hom}_{R}(R, M)$ into a left $R$ module by defining $r f$ (for $f: R \rightarrow M$ and $r \in R$ ) by $r f: s \mapsto f(s r)$ for all $s \in R .$
(ii) If $g: M \rightarrow N$, prove that the following diagram commutes:
Conclude that $\varphi=\left(\varphi_{M}\right)_{M \in o b j(R M o d)}$ is a natural isomorphism from $\operatorname{Hom}_{R}(R, \square)$ to the identity functor on ${ }_{R} \mathbf{M o d}$.
[Compare with Example 1.16(ii).]

Key Concepts

Functor

A functor is a mapping between categories that associates objects to objects and morphisms to morphisms in such a way that the composition of morphisms and identities are preserved. Functors provide a framework to systematically translate concepts across different categories, which is essential for the abstract study of algebraic structures.

Natural Transformation

A natural transformation is a collection of morphisms that provides a way of transforming one functor into another while respecting the structure of the categories involved. The concept encapsulates the idea of 'naturality' in comparisons of functors and is crucial in understanding when two constructions are, in an essential way, the same, thus leading to the notion of natural isomorphisms.

Isomorphism

An isomorphism is a bijective map that not only establishes a one-to-one correspondence between elements of two structures but also preserves the underlying operations. In the context of modules, showing that an evaluation map is an isomorphism confirms that the two modules in question are fundamentally the same in structure, thus allowing one to replace one with the other in theoretical developments.

Representable Functor

A representable functor is one that is naturally isomorphic to a Hom-functor, meaning its behavior can be fully captured by the homomorphisms from a fixed object. This idea is foundational in many areas of category theory, including the Yoneda lemma, and provides deep insights into the connection between abstract categorical constructions and concrete algebraic structures.

Module Homomorphisms

Module homomorphisms are structure-preserving maps between modules that maintain the operations of addition and scalar multiplication. These maps are central to understanding how modules relate to each other in a way that keeps the module structure intact, forming the basis for many constructions and results in module and category theory.

Left R-Modules

A left R-module is an algebraic structure where the elements of a ring R 'act' linearly on an abelian group. This concept generalizes vector spaces by allowing the scalars to come from a ring instead of a field, and it is fundamental to module theory, which studies structures and homomorphisms that respect this action.

Evaluation Map

The evaluation map is a function that 'extracts' the value of a module homomorphism at a particular element from the domain, often used to convert a function into a scalar or an element of the codomain. It plays a critical role in establishing isomorphisms between functorially defined objects and given modules by exploiting the properties of the domain, such as the identity element in a ring.

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