Prove that every left exact covariant functor T: R Mod →Ab preserves pullbacks. Conclude that if

step 1 And this question we're going to be dealing with subsets and intersections. So we've been asked

Prove that every left exact covariant functor T: R Mod →Ab...

Prove that every left exact covariant functor $T: R$ Mod $\rightarrow$ Ab preserves pullbacks. Conclude that if $B$ and $C$ are submodules of a module $A$, then for every module $M$, we have $$ \operatorname{Hom}_{R}(M, B \cap C)=\operatorname{Hom}_{R}(M, B) \cap \operatorname{Hom}_{R}(M, C) . $$

Prove that every left exact covariant functor $T: R$ Mod $\rightarrow$ Ab preserves pullbacks. Conclude that if $B$ and $C$ are submodules of a module $A$, then for every module $M$, we have
$$
\operatorname{Hom}_{R}(M, B \cap C)=\operatorname{Hom}_{R}(M, B) \cap \operatorname{Hom}_{R}(M, C) .
$$

Key Concepts

Submodule Intersection

In module theory, the intersection of submodules represents the elements that are common to both submodules. When dealing with the Hom functor, the property that Hom(M, B ? C) = Hom(M, B) ? Hom(M, C) reflects the idea that applying a left exact functor preserves this intersection structure. This equivalence illustrates how algebraic operations on submodules transfer to intersections of sets of homomorphisms.

Hom Functor

The Hom functor assigns to a pair of modules the abelian group of module homomorphisms between them. It plays a central role in module theory and homological algebra, especially when relating structural properties of modules. In the problem, the Hom functor’s interaction with pullbacks explains why taking the Hom of an intersection (or pullback) yields the intersection of the Hom sets, reinforcing the idea that certain limit constructions are preserved.

Left Exact Functor

A left exact functor is one that preserves all finite limits, particularly kernels and finite products. In the context of module categories, this means that when a functor is applied to an exact sequence at the beginning, the exactness at that part of the sequence is maintained. This property is crucial in homological algebra because it ensures that essential structures like kernels and pullbacks (which are types of limits) are preserved under the functor.

Pullback

The pullback is a fundamental construction in category theory that generalizes the notion of an intersection or a fibered product. It is defined as a universal object that maps to two objects in a way that their images coincide when composed with corresponding maps to a third object. The preservation of pullbacks by a functor ensures that the relationships expressed by these universal constructions are maintained when the functor is applied.

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