Suppose one assumes, in the hypothesis of Proposition 2.42,...

Suppose one assumes, in the hypothesis of Proposition 2.42, that the induced map $i^{*}: \operatorname{Hom}_{R}(B, M) \rightarrow \operatorname{Hom}_{R}\left(B^{\prime}, M\right)$ is surjective for every $M$. Prove that $0 \rightarrow B^{\prime} \stackrel{i}{\longrightarrow} B \stackrel{p}{\longrightarrow} B^{\prime \prime} \rightarrow 0$ is a split short exact sequence.

Key Concepts

Short Exact Sequence

A short exact sequence of modules is a sequence 0 ? A ? B ? C ? 0 where the image of each map is exactly the kernel of the following map. This means that A embeds into B, C is the quotient of B by the image of A, and there are no “gaps” or redundancies in the mapping. This concept is fundamental in homological algebra and module theory as it encapsulates both embedding and surjectivity properties in a concise exact formulation.

Split Exact Sequence

A split exact sequence is a special type of short exact sequence where the module in the middle is isomorphic to the direct sum of the modules at the ends. In other words, the sequence 0 ? A ? B ? C ? 0 splits if there exists a module homomorphism from C back into B (or from B to A) that serves as a right- or left-inverse to the projection or inclusion map, respectively. This splitting implies a decomposition of B into simpler, more manageable pieces, which is a powerful tool in module theory and homological algebra.

Hom Functor

The Hom functor, denoted Hom_R(–, M) for a fixed module M, is a fundamental construction in module theory that assigns to each module the set of all R-linear maps from that module to M. This functor is covariant or contravariant depending on the argument and plays a central role in understanding how module homomorphisms interact with sequences. It preserves or reflects certain exactness properties under additional assumptions, making it essential for studying extensions, lifting properties, and splitting conditions in exact sequences.

Lifting Property in Module Theory

The lifting property in module theory refers to the ability to extend or 'lift' a homomorphism defined on a submodule to the entire module. In the context of exact sequences, if every homomorphism defined on the submodule can be extended to a homomorphism on the full module, it often indicates that the sequence splits. This property, when phrased in terms of the surjectivity of induced Hom maps, provides a powerful criterion for proving that a short exact sequence is split, thereby revealing structural information about the modules involved.

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