Show that the direct sum of a flat module and a faithfully...

Show that the direct sum of a flat module and a faithfully flat module is faithfully flat. Conclude, therefore, that any nonzero free module is faithfully flat. Find an example of a flat $\mathbb{Z}$-module that is not faithfully flat.

Key Concepts

Flat Module

A flat module is one for which the tensor product with any exact sequence of modules remains exact. This means that the process of tensoring by a flat module preserves the structure of modules and their relationships, ensuring that no 'new' torsion is introduced. Flatness is a crucial concept in module theory and homological algebra, particularly when studying the behavior of modules under base change.

Faithfully Flat Module

A faithfully flat module is a flat module that in addition reflects exactness; that is, if tensoring by the module sends a sequence to an exact sequence, then the original sequence was already exact. This property ensures that the module 'detects' nontrivial information about the ring, and thus faithfully flat modules are used in descent theory and in understanding extensions and localization in algebra.

Direct Sum

The direct sum of modules is a construction that combines modules in such a way that each element is expressed uniquely as a finite sum of elements from each module. In the context of flatness, the direct sum of modules tends to preserve the flatness property. When one of the summands is faithfully flat, under appropriate conditions, the whole direct sum inherits the property of being faithfully flat, ensuring that the larger module still reflects the essential features of the ring.

Free Module

A free module is one that has a basis, meaning it is isomorphic to a direct sum of copies of the ring, and every element can be uniquely represented as a finite linear combination of these basis elements. Free modules are important in algebra because they have a simple and explicit structure. Moreover, any nonzero free module is faithfully flat since it contains at least one copy of the ring, ensuring that it reflects the ring’s structure accurately under tensoring.

Example of a Flat but not Faithfully Flat Module

A classic example in the context of ?-modules is the field of rational numbers, ?. As a ?-module, ? is flat because tensoring with ? preserves the exactness of sequences, taking advantage of its divisibility and lack of torsion. However, ? is not faithfully flat over ? since there are nonzero ?-modules, such as the finite cyclic groups, whose structure is annihilated when tensored with ?. This explains why ?, despite being flat, does not detect all non-trivial information about ?-modules.

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