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Tag 01DA

Chapter 15: More on Algebra > Section 15.51: Injective modules

Lemma 15.51.6. Let $R$ be a ring. The functor $M \mapsto M^\vee$ is exact.

Proof. This because $\mathbf{Q}/\mathbf{Z}$ is an injective abelian group by Lemma 15.50.1. $\square$

    The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 11884–11888 (see updates for more information).

    \begin{lemma}
    \label{lemma-vee-exact}
    Let $R$ be a ring.
    The functor $M \mapsto M^\vee$ is exact.
    \end{lemma}
    
    \begin{proof}
    This because $\mathbf{Q}/\mathbf{Z}$
    is an injective abelian group by Lemma \ref{lemma-injective-abelian}.
    \end{proof}

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