Some fixes

Signed-off-by: zeramorphic <[email protected]>

ib/grm/09_factorisation.tex

@@ -132,7 +132,7 @@ \subsection{Principal ideal domains}
 	Note that this direction of the proof did not require that \( R \) was a principal ideal domain, however \( R \) must still be an integral domain.
 \end{proof}
 \begin{remark}
-	Let \( R \) be a principal integral domain, and \( 0 \neq r \in R \).
+	Let \( R \) be a principal ideal domain, and \( 0 \neq r \in R \).
 	Then, \( (r) \) is maximal if and only if \( r \) is irreducible, which is true if and only if \( r \) is prime, which is equivalent to the fact that \( (r) \) is prime.
 	Hence, the maximal ideals are the nonzero prime ideals.
 \end{remark}

iii/commalg/01_chain_conditions.tex

@@ -95,7 +95,7 @@ \subsection{Noetherian and Artinian modules}
     \begin{enumerate}
         \item \( \mathbb Z \) over itself is a Noetherian module as it is a principal ideal domain, but it is not an Artinian module because we can take the chain \( (2) \supsetneq (4) \supsetneq (8) \supsetneq \cdots \).
         \item \( \mathbb Z \) is similarly a Noetherian ring but not an Artinian ring by unfolding the definition and using (i).
-        \item \( \faktor{\mathbb Z\qty[\frac{1}{2}]\,}{\mathbb Z} \) is an Artinian module but not a Noetherian module.
+        \item \( \faktor{\mathbb Z\qty[\frac{1}{2}]\,}{\mathbb Z} \) is an Artinian \( \mathbb Z \)-module but not a Noetherian \( \mathbb Z \)-module. This can be seen from the fact that the only submodules are of the form \( \qty(\frac{1}{2^k} + \mathbb Z) \) for \( k \in \mathbb N \).
         \item In fact, a ring \( R \) is Artinian if and only if \( R \) is Noetherian and \( R \) has \emph{Krull dimension} 0.
     \end{enumerate}
 \end{example}

iii/commalg/03_localisation.tex

@@ -321,9 +321,9 @@ \subsection{Extension and contraction of ideals}
 In this case, the extension of an ideal is written \( S^{-1}\mathfrak a = \mathfrak a^e \).
 We claim that
 \[ \mathfrak a^e = \qty{\frac{a}{s} \midd a \in \mathfrak a, s \in S} \]
-Indeed, \( \mathcal a^e \) is generated by \( \qty{\frac{a}{1} \midd a \in \mathfrak a} \), so \( \mathfrak a^e \) must contain \( \qty{\frac{a}{s} \midd a \in \mathfrak a, s \in S} \), but this is already an ideal.
+Indeed, \( \mathfrak a^e \) is generated by \( \qty{\frac{a}{1} \midd a \in \mathfrak a} \), so \( \mathfrak a^e \) must contain \( \qty{\frac{a}{s} \midd a \in \mathfrak a, s \in S} \), but this is already an ideal.
 We also claim that
-\[ \mathcal a^{ec} = \bigcup_{s \in S} (\mathfrak a : s);\quad (\mathfrak a : s) = \qty{r \in R \mid rs \in \mathfrak a} \]
+\[ \mathfrak a^{ec} = \bigcup_{s \in S} (\mathfrak a : s);\quad (\mathfrak a : s) = \qty{r \in R \mid rs \in \mathfrak a} \]
 Indeed, for \( r \in \bigcup_{s \in S} (\mathfrak a : s) \), we have \( rs = a \) in \( R \) for some \( s \in S \) and \( a \in \mathfrak a \), so \( \frac{rs}{1} = \frac{a}{1} \), giving \( \frac{r}{1} = \frac{a}{s} \), so \( r \in \mathfrak a^{ec} \) as required.
 In the other direction, if \( r \in \mathfrak a^{ec} \), then \( \frac{r}{1} = \frac{a}{s} \) for some \( s \in S \) and \( a \in \mathfrak a \), so there exists \( u \in S \) such that \( rus = ua \in \mathfrak a \), so \( r \in (\mathfrak a : us) \) as required.
 

iii/lc/03_reflection.tex

@@ -121,7 +121,7 @@ \subsection{Ultrapowers of the universe}
         \item If \( f : \kappa \to \mathrm{V}_\lambda \) is arbitrary, the set
         \[ \qty{\rank f(\alpha) \mid \alpha \in \kappa} \]
         cannot be cofinal in \( \lambda \), so there is \( \gamma < \lambda \) such that \( f \in \mathrm{V}_\gamma \).
-        However, the equivalence class \( [f] \) is unbounded in \( \mathrm{V}_\lambda \).
+        However, the union of the equivalence class \( [f] \) is unbounded in \( \mathrm{V}_\lambda \).
         \item Given \( f \), by (ii) we may assume \( f \in \mathrm{V}_\gamma \) for some \( \gamma < \lambda \).
         If \( [g] \mathrel{E} [f] \), then
         \[ X = \qty{\alpha \mid g(\alpha) \in f(\alpha)} \in U \]
@@ -217,7 +217,7 @@ \subsection{Ultrapowers of the universe}
     \[ \alpha < (\id_\kappa) < j(\kappa) \]
     giving
     \[ \kappa \leq (\id_\kappa) < j(\kappa) \]
-    as required
+    as required.
 \end{proof}
 \begin{remark}
     \begin{enumerate}
@@ -241,6 +241,8 @@ \subsection{Ultrapowers of the universe}
         We will discuss this in more detail later.
     \end{enumerate}
 \end{remark}
+
+\subsection{Properties above the critical point}
 \begin{definition}
     Let \( j : \mathrm{V}_\lambda \to M \) be an elementary embedding such that \( M \subseteq \mathrm{V}_\lambda \) is transitive.
     An ordinal \( \mu \) is called the \emph{critical point} of \( j \), written \( \operatorname{crit}(j) \), if \( j \neq \id \) and \( \mu \) is the least ordinal \( \alpha \). such that \( j(\alpha) > \alpha \).
@@ -307,14 +309,14 @@ \subsection{Ultrapowers of the universe}
 There could be some other \( U' \in V_{\kappa + 2} \) which is \( \kappa \)-complete and nonprincipal.
 
 Recall that the Keisler extension property for a transitive model \( X \) is the statement that there is \( \kappa \in X \) such that \( \mathrm{V}_\kappa \preceq X \).
-Properties of \( X \) reflect down into \( \mathrm{V}_\kappa \): if \( \alpha \in \mathrm{Ord}^X \) and \( \Phi \) is a property such that \( X \vDash \Phi(\kappa) \), then
-\[ M \vDash \exists \mu.\, \alpha < \mu \wedge \Phi(\mu) \]
+Properties of \( X \) reflect down into \( \mathrm{V}_\kappa \): if \( \alpha \in \mathrm{Ord}^{\mathrm{V}_\kappa} \) and \( \Phi \) is a property such that \( X \vDash \Phi(\kappa) \), then
+\[ X \vDash \exists \mu.\, \alpha < \mu \wedge \Phi(\mu) \]
 so
 \[ \mathrm{V}_\kappa \vDash \exists \mu.\, \alpha < \mu \wedge \Phi(\mu) \]
 hence
 \[ C_\Phi = \qty{\gamma < \kappa \mid \Phi(\gamma)} \subseteq \kappa \]
 is cofinal in \( \kappa \).
-If \( \Phi \) is any property such that \( M \vDash \Phi(\kappa) \), then for any \( \alpha < \kappa \),
+Now, if \( \Phi \) is any property such that \( M \vDash \Phi(\kappa) \), then for any \( \alpha < \kappa \),
 \[ M \vDash \exists \mu.\, j(\alpha) < \mu < j(\kappa) \wedge \Phi(\mu) \]
 By elementarity,
 \[ \mathrm{V}_\lambda \vDash \exists \mu.\, \alpha < \mu < \kappa \wedge \Phi(\mu) \]