Exercise 111.2.2. Let $I$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$. Show that there exists a ring $A$ and maps $\varphi _ i : A_ i \to A$ such that $\varphi _ j \circ \varphi _{ij} = \varphi _ i$ for all $i \leq j$ with the following universal property: Given any ring $B$ and maps $\psi _ i : A_ i \to B$ such that $\psi _ j \circ \varphi _{ij} = \psi _ i$ for all $i \leq j$, then there exists a unique ring map $\psi : A \to B$ such that $\psi _ i = \psi \circ \varphi _ i$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)