Prove that an exact sequence 0 →B^' →B →B^'' →0 of left R modules is pure exact if and only if i

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Prove that an exact sequence 0 →B^' →B →B^'' →0 of left R...

Prove that an exact sequence $0 \rightarrow B^{\prime} \rightarrow B \rightarrow B^{\prime \prime} \rightarrow 0$ of left $R$ modules is pure exact if and only if it remains exact after tensoring by all finitely presented right $R$-modules $A$. Hint. That an element lies in $\operatorname{ker}\left(A \otimes_{R} B^{\prime} \rightarrow A \otimes_{R} B\right)$ involves only finitely many elements of $A$.

Key Concepts

Finitely Presented Module

A finitely presented module is defined by a presentation with a finite number of generators and relations, meaning that its structure can be completely described by a finite amount of data. This property is important in various areas of algebra, including the analysis of tensor products, because the finite generation condition ensures that many constructions and proofs, such as those involving commuting limits with tensor product, behave well.

Tensor Product of Modules

The tensor product is an operation that allows one to combine two modules over a ring to form a new module, capturing bilinear relationships between elements. In the context of exact sequences, tensor products are used to transfer structures and determine how module operations interact under change of scalars, making them a central tool in the investigation of properties like purity in module theory.

Exact Sequence

An exact sequence in module theory is a sequence of module homomorphisms such that the image of one homomorphism is precisely the kernel of the next. This property ensures that the sequence 'fits together' perfectly by balancing how elements map between modules, and it is a fundamental concept in homological algebra for understanding the structure and relationships between modules.

Pure Exact Sequence

A pure exact sequence is an exact sequence of modules that remains exact after tensoring with any module. This stronger notion of exactness is significant because it reflects a kind of compatibility of the sequence with all module structures, ensuring that the property of being a direct summand in a larger module persists through tensor products, which is a crucial aspect in the study of module purity and decompositions.

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