Proposition 2

Let $(E_\alpha, f_{\beta \alpha})$ be a direct system of magmas and let $E$ be its direct limits $f_\alpha : E_\alpha \to E$ the canonical homomorphisms. Suppose a magma $F$ and homomorphisms $u_\alpha : E_\alpha \to F$ are given such that $u_\alpha = u_\beta \circ f_{\beta \alpha}$ for $\alpha \leq \beta$. There exists one and only one homomorphism $u : E \to F$ such that $u_\alpha = u \circ f_\alpha$ for all $\alpha \in I$. If the magmas $E_\alpha$ and $F$ and the homomorphisms $f_{\beta \alpha}$ and $u_\alpha$ are unital, the homomorphism $u$ is unital.

We know (Set Theory, III, § 7, no. 6, Proposition 6) that there exists one and only one mapping $u : E \to F$ such that $u_\alpha = u \circ f_\alpha$ for all $\alpha \in I$. We verify that $u$ is a homomorphism: let $x, y$ be in $E$, $\alpha$ in $I$ and $x_\alpha, y_\alpha$ in $E_\alpha$ such that $x = f_\alpha(x_\alpha)$ and $y = f_\alpha(y_\alpha)$. Then $xy = f_\alpha(x_\alpha y_\alpha)$, whence

We consider now the unital case and let $e_\alpha$ denote the identity element of $E_\alpha$, $e$ that of $E$ and $e'$ that of $F$. Choose $\alpha \in I$, then $e = f_\alpha(e_\alpha)$, whence for $u_\alpha$ is unital. Hence $u$ is unital.

By analogy with the notion of direct system of magmas, that of a direct system of monoids or groups can be formulated. Proposition 1 shows that the magma E which is the limit of a direct system of monoids $(E_\alpha, f_{\beta\alpha})_{\alpha, \beta \in I}$ is a monoid. We show that E is a group if the $E_\alpha$ are groups: let $x \in E, \alpha \in I$ and $x_\alpha \in E_\alpha$ be such that $x = f_\alpha(x_\alpha)$; the element $y = f_\alpha(x_\alpha^{-1})$ of E is the inverse of x (§ 2, no. 3). The universal property of Proposition 2 goes over immediately in the case of a direct system of monoids or groups.

The reader is left to define a direct system of rings. Let $(A_\alpha, f_{\beta\alpha})$ be such a direct system; let $A = \lim \overrightarrow{A_\alpha}$ and $f_\alpha : A_\alpha \to A$ the canonical homomorphisms. There exists (Proposition 2) on A one and only one addition and multiplication characterized by $x + y = f_\alpha(x_\alpha + y_\alpha), xy = f_\alpha(x_\alpha y_\alpha)$ for $\alpha \in I, x_\alpha, y_\alpha$ in $A_\alpha$ and $x = f_\alpha(x_\alpha), y = f_\alpha(y_\alpha)$. Under addition A is a commutative group and multiplication is associative and unital. Finally, for $x, y, z$ in A, choose $\alpha$ in I and $x_\alpha, y_\alpha$ and $z_\alpha$ in $A_\alpha$ with

Then and the relation $x(y + z) = xy + xz$ is proved analogously. In other words, A has the structure of a ring, characterized by the fact that $f_\alpha$ is a ring homomorphism for all $\alpha \in I$.

The ring A is called the direct limit of the rings $A_\alpha$. Proposition 2 extends immediately to the case of rings.