Exercise 2.5

Question

Let $A[x]$ be the ring of polynomials in one indeterminate over a ring $A$. Prove that $A[x]$ is a flat $A$-algebra.

Amit Awasthi · Accepted Answer

\begin{proof}
  As an $A$-module, $A[x] \cong A \oplus xA \oplus x^2A \oplus \cdots \cong A \oplus A \oplus A \oplus \cdots$. Since tensoring by $A$ is trivial by Prop.~$2.14iv)$, $A$ is flat, so $A[x]$ is flat by Exercise 2.4.
\end{proof}

Exercise 2.5

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Exercise 2.5

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