describe epimorphisms of commutative algebras

Author: ScriptRaccoon

database/data/007_special-morphisms/004_epimorphisms.sql

@@ -70,6 +70,11 @@ VALUES
 	'A functor $F : \mathcal{C} \to \mathcal{D}$ is an epimorphism iff $F$ is surjective on objects and for every morphism $s$ in $\mathcal{D}$ there is a zigzag over $U := F(\mathcal{C})$, meaning morphisms $u_1,\dotsc,u_{m+1} \in U$, $v_1,\dotsc,v_m \in U$, $x_1,\dotsc,x_m \in \mathcal{D}$ and $y_1,\dotsc,y_m \in \mathcal{D}$ such that $s = x_1 u_1$, $u_1 = v_1 y_1$, $x_{i-1} v_{i-1} = x_i u_i$, $u_i y_{i-1} = v_i y_i$, $x_m v_m = u_{m+1}$ and $u_{m+1} y_m = s$.',
 	'This is an extension of the <a href="https://en.wikipedia.org/wiki/Isbell''s_zigzag_theorem" target="_blank">corresponding theorem for monoids</a> and proven in <a href="https://www.jstor.org/stable/2373286" target="_blank">Epimorphisms and Dominions, III</a> by John R. Isbell.'
 ),
+(
+	'CAlg(R)',
+	'a homomorphism of algebras which is an epimorphism of commutative rings',
+	'The forgetful functor $\mathbf{CAlg}(R) \to \mathbf{Ring}$ is faithful and hence reflects epimorphisms, but it also preserves epimorphisms since it preserves pushouts (since $\mathbf{CAlg}(R) \cong R / \mathbf{Ring}$). For epimorphisms of commutative rings see their <a href="/category/CRing">detail page</a>.'
+),
 (
 	'CRing',
 	'A ring map $f : R \to S$ is an epimorphism iff $S$ equals the <i>dominion</i> of $f(R) \subseteq S$, meaning that for every $s \in S$ there is some matrix factorization $(s) = Y X Z$ with $X \in M_{n \times n}(R)$, $Y \in M_{1 \times n}(S)$, and $Z \in M_{n \times 1}(S)$.',