# The Stacks Project

## Tag 058I

Definition 10.81.1. Let $f: M \to N$ be a map of $R$-modules. Then $f$ is called universally injective if for every $R$-module $Q$, the map $f \otimes_R \text{id}_Q: M \otimes_R Q \to N \otimes_R Q$ is injective. A sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ of $R$-modules is called universally exact if it is exact and $M_1 \to M_2$ is universally injective.

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 18700–18708 (see updates for more information).

\begin{definition}
\label{definition-universally-injective}
Let $f: M \to N$ be a map of $R$-modules.  Then $f$ is called
{\it universally injective} if for every $R$-module $Q$, the map $f \otimes_R \text{id}_Q: M \otimes_R Q \to N \otimes_R Q$
is injective.  A sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ of $R$-modules is called {\it universally exact} if it is exact
and $M_1 \to M_2$ is universally injective.
\end{definition}

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