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The following remark is given in the book:

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And the excercise that is referred to is given in the following picture:

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And the solution to the excercise is given in the following pictures:

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I do not understand what is the concrete category that all R-modules and R-module homomorphisms forms?

Also why is the definitions of monomorphisms in (a) and epimorphisms in (b) are strictly in categorical terms?

Could anyone explain this for me please?

Thanks!

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In the second question, you must show that the usual definitions of epimorphism and monomorphism are equivalents to their categorical versions. When they say the concrete category of R-modules and homomorphism of R-modules the are talking about the category whose objets are modules over a fixed ring R and the morphism are homomorphism between these R modules which is a concrete category. In wikipedia you can see the following definition:

A $\textbf{concrete category}$ is a pair $(\mathcal{C},U)$ such that:

  1. $\mathcal{C}$ is a category,

  2. $U:\mathcal{C}\longrightarrow\rm{\bf Sets}$ is a ${\bf faithful}$ functor.

Examples: The category of Groups ${\bf Gr}$ and homomorphism of groups is concrete. The same for $Ab$ and for a unitary commutative ring $R$ the categories R-mod or mod-R of left or right R-modules.

It is useful to know that something similar to this can be proved in the category Sets also. Is good to start by the notion of sections and retractions in the category Sets and after in the category R-mod.

For instance, in the category $\rm{\bf Sets}$ you have the following result:

Let $A$,$B$ be two sets and $\ f:A\longrightarrow B\ $ be a map. Then,

  1. The map $\ f\ $ is surjective if and only if for every pair of maps $g_{1}:B\longrightarrow C$ and $g_{2}:B\longrightarrow C$ and all set $C$ we have: $$ g_{1}\circ f=g_{2}\circ f \quad\ \Longrightarrow\ \quad g_{1}=g_{2}. $$

  2. The map $\ f\ $ is injective if and only if for every pair of maps $\ h_{1}:C\longrightarrow A\ $ and $\ h_{2}:C\longrightarrow A\ $ and a set $C$ we have: $$ f\circ h_{1}=f\circ h_{2}\quad\ \Longrightarrow\ \quad h_{1}=h_{2}.$$

So, this implies that in the category $\ \rm{\bf Sets}\ $ you can define the notion of injective map and the notion surjective map strictly in categorical terms, that is, you can define these notions without speaking about the elements of $A$ or $B$ like I wrote above.

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