Let 0 →A →B →C →0 be a short exact sequence of left...

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of left $R$-modules. If $M$ is any left $R$-module, prove that there are exact sequences $$ 0 \rightarrow A \oplus M \rightarrow B \oplus M \rightarrow C \rightarrow 0 $$ and $$ 0 \rightarrow A \rightarrow B \oplus M \rightarrow C \oplus M \rightarrow 0 $$

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of left $R$-modules. If $M$ is any left $R$-module, prove that there are exact sequences
$$
0 \rightarrow A \oplus M \rightarrow B \oplus M \rightarrow C \rightarrow 0
$$
and
$$
0 \rightarrow A \rightarrow B \oplus M \rightarrow C \oplus M \rightarrow 0
$$

Key Concepts

Direct Sum of Modules

The direct sum of modules, denoted A ? M, represents a construction that combines two modules into a new module where operations are performed coordinate-wise. This construction is essential in module theory since it allows one to express modules as sums of simpler modules; in particular, it provides a way to 'add' an arbitrary module M to other modules in an exact manner, often preserving or revealing additional structural properties.

Additive Functors and Exactness

An additive functor is a functor that preserves the structure of abelian groups, including finite direct sums and zero objects. The direct sum operation is an example of a functor that, when applied to exact sequences, often preserves their exactness due to its additive nature. This concept is critical for understanding why taking direct sums with a fixed module, as in the given problem, results in new sequences that remain exact.

Short Exact Sequence

A short exact sequence is an exact sequence of the form 0 ? A ? B ? C ? 0, where the map from A to B is injective, the map from B to C is surjective, and the image of the first map is exactly the kernel of the second map. This concept is fundamental in homological algebra and module theory because it describes how a module B can be 'built' from a submodule A and a quotient module C, leading to many structural and extension questions about modules.

Module Homomorphisms and Their Properties

Module homomorphisms are structure-preserving maps between modules that respect the operations of addition and scalar multiplication. In the context of exact sequences, the properties of module homomorphisms—such as injectivity, surjectivity, and the relationship between images and kernels—are key. These properties ensure that when one extends a sequence by adding a direct summand, one can maintain the exactness of the sequence by carefully defining the corresponding maps.

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