A right R-module B is called faithfully flat if (i) B is a...

A right $R$-module $B$ is called faithfully flat if (i) $B$ is a flat module, (ii) for all left $R$-modules $X$, if $B \otimes_{R} X=\{0\}$, then $X=\{0\}$. Prove that $R[x]$ is a faithfully flat $R$-module (if $R$ is not commutative, then $R[x]$ is the polynomial ring in which the indeterminate $x$ commutes with each coefficient in $R$ ).

A right $R$-module $B$ is called faithfully flat if
(i) $B$ is a flat module,
(ii) for all left $R$-modules $X$, if $B \otimes_{R} X=\{0\}$, then $X=\{0\}$.
Prove that $R[x]$ is a faithfully flat $R$-module (if $R$ is not commutative, then $R[x]$ is the polynomial ring in which the indeterminate $x$ commutes with each coefficient in $R$ ).

Key Concepts

Flat Module

A flat module is one that preserves the exactness of sequences when tensored with any module. In other words, if you have a short exact sequence of modules, then after tensoring with a flat module, the resulting sequence remains exact. This concept is central in homological algebra and ensures that the functor of tensoring with a flat module is exact.

Faithfully Flat Module

A faithfully flat module is a flat module that also detects nonzero modules. That is, if the tensor product of a faithfully flat module with any other module yields the zero module, then that other module must itself be zero. This extra condition is important for various applications, such as descent theory and base change, because it guarantees that tensoring does not lose any information about the module structure.

Tensor Product of Modules

The tensor product is a construction that allows one to combine modules over a ring to form another module. It is a bilinear operation that encodes how modules interact with one another over the ring. The behavior of the tensor product, particularly its exactness properties with flat modules, plays a crucial role in module theory and homological algebra.

Polynomial Ring Extensions

A polynomial ring extension is formed by adjoining an indeterminate to a ring, resulting in a new ring of polynomials. When viewing a polynomial ring as a module over the original ring, it often exhibits good properties such as flatness. In the context of faithfully flat modules, showing that a polynomial ring extension does not annihilate any nonzero module when tensored with it is a key aspect of proving its faithful flatness.

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