## Tag `058I`

Chapter 10: Commutative Algebra > Section 10.81: Universally injective module maps

Definition 10.81.1. Let $f: M \to N$ be a map of $R$-modules. Then $f$ is called

universally injectiveif 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 calleduniversally exactif 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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