Let 0 →A →B →C →0 be an exact sequence of right R modules,...

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an exact sequence of right $R$ modules, for some ring $R$. If both $A$ and $C$ are flat modules, prove that $B$ is a flat module.
Hint. This result is routine if one uses the derived functor Tor.

Key Concepts

Exact Sequence

An exact sequence is a chain of module homomorphisms where the image of one map coincides with the kernel of the subsequent map. This concept is essential because it allows the decomposition of complex structures into simpler parts and is widely used in homological algebra to study and transfer properties between modules.

Flat Module

A flat module is one that, when tensored with any other module, preserves the exactness of sequences. This means that if you have an exact sequence and you tensor it with a flat module, the resulting sequence remains exact. Flatness is an important property that ensures good behavior with respect to tensor operations and is central to many constructions in module theory and homological algebra.

Derived Functor Tor

The Tor functor is a tool in homological algebra that arises as a derived functor of the tensor product. It measures the extent to which the tensor product fails to be exact. In particular, a module is flat if and only if the first Tor functor vanishes for all modules, making Tor a critical instrument in testing and proving the flatness property.

Tensor Functor

The tensor functor takes a pair of modules and produces another module, usually not preserving exactness on the left by default. However, when one of the modules is flat, tensoring becomes an exact operation. This relationship between the tensor functor and flat modules is a key point in understanding how flatness interacts with module homomorphisms and exact sequences.

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