(i) Prove that a right R-module B is flat if and only if...

(i) Prove that a right $R$-module $B$ is flat if and only if exactness of any sequence of left $R$-modules $A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{P}{\longrightarrow} A^{\prime \prime}$ implies exactness of $B \otimes R A^{\prime} \stackrel{1 \otimes i}{\longrightarrow} B \otimes R A \stackrel{1 \otimes p}{\longrightarrow} B \otimes R A^{\prime \prime}$. (ii) Prove that a right $R$-module $B$ is faithfully flat if and only if it is flat and $B \otimes_{R} A^{\prime} \stackrel{1 \otimes i}{\longrightarrow} B \otimes_{R} A \stackrel{1 \otimes p}{\longrightarrow} B \otimes_{R} A^{\prime \prime}$ exact implies $A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} A^{\prime \prime}$ is exact.

(i) Prove that a right $R$-module $B$ is flat if and only if exactness of any sequence of left $R$-modules $A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{P}{\longrightarrow} A^{\prime \prime}$ implies exactness of $B \otimes R A^{\prime} \stackrel{1 \otimes i}{\longrightarrow} B \otimes R A \stackrel{1 \otimes p}{\longrightarrow} B \otimes R A^{\prime \prime}$.
(ii) Prove that a right $R$-module $B$ is faithfully flat if and only if it is flat and $B \otimes_{R} A^{\prime} \stackrel{1 \otimes i}{\longrightarrow} B \otimes_{R} A \stackrel{1 \otimes p}{\longrightarrow} B \otimes_{R} A^{\prime \prime}$ exact implies $A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} A^{\prime \prime}$ is exact.

Key Concepts

Flat Module

A flat module is one for which the tensor product with any exact sequence of modules preserves exactness. This concept is fundamental in homological algebra as it ensures that tensoring with a flat module does not introduce any new torsion or distort the structure of exact sequences, thereby allowing one to reliably transfer properties from one module to another via the tensor product functor.

Exact Sequence

An exact sequence is a sequence of module homomorphisms where the image of one map coincides with the kernel of the subsequent map. This notion is vital for understanding the structure and decomposition of modules, and in homological algebra it serves as the backbone for many constructions and proofs, such as those involving flat and faithfully flat modules.

Tensor Product

The tensor product is an operation that constructs a new module from a pair of modules over a ring, effectively 'multiplying' the modules together in a bilinear fashion. In the context of flatness, it is crucial because the property of being flat is characterized by the exactness of the tensor product functor; that is, if tensoring with a module preserves the exactness of any sequence, then the module is flat.

Faithfully Flat Module

A faithfully flat module is one that is not only flat—meaning that tensoring with it preserves the exactness of sequences—but also reflects exactness. That is, if tensoring a sequence with the module results in an exact sequence, then the original sequence must have been exact. Faithfully flat modules play a key role in descent theory and other areas of algebra by ensuring that essential properties are maintained and can be recovered from the tensor product.

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