replace \fM

2021_Algebra_I.tex

@@ -53,7 +53,7 @@ Fields which are not assumed to be algebraically closed have been renamed (usual
     \begin{enumerate}
         \item Every submodule  $N \subseteq M$ is finitely generated.
         \item Every sequence $N_0 \subset N_1 \subset \ldots$ of submodules terminates
-        \item Every set $\fM \neq \emptyset$ of submodules of $M$ has a $\subseteq$-largest element.
+        \item Every set $\mathfrak{M} \neq \emptyset$ of submodules of $M$ has a $\subseteq$-largest element.
     \end{enumerate}
 \end{definition}
 \begin{proposition}[Hilbert's Basissatz]\label{basissatz}
@@ -640,8 +640,8 @@ Let $X$ be a topological space.
 
 \begin{proof}
     % i = ic
-    Let $\fM$ be the set of closed subsets of $X$ which cannot be decomposed as a union of finitely many irreducible subsets.
-    Suppose $\fM \neq  \emptyset$. Then there exists a $\subseteq$-minimal $Y \in \fM$. $Y$ cannot be empty or irreducible. Hence  $Y = A \cup  B$ where $A,B$ are proper closed subsets of $ Y$. By the minimality of $Y$, $A$ and $B$ can be written as a union of proper closed subsets  $\lightning$.
+    Let $\mathfrak{M}$ be the set of closed subsets of $X$ which cannot be decomposed as a union of finitely many irreducible subsets.
+    Suppose $\mathfrak{M} \neq  \emptyset$. Then there exists a $\subseteq$-minimal $Y \in \mathfrak{M}$. $Y$ cannot be empty or irreducible. Hence  $Y = A \cup  B$ where $A,B$ are proper closed subsets of $ Y$. By the minimality of $Y$, $A$ and $B$ can be written as a union of proper closed subsets  $\lightning$.
 
     Let $X = \bigcup_{i = 1}^n Z_i$, where there are no inclusions between the $Z_i$. If $Y$ is an irreducible subsets of $X$, $Y = \bigcup_{i = 1}^n (Y \cap Z_i)$ and there exists $1 \le i \le n$ such that $Y = Y \cap Z_i$.
     Hence $Y \subseteq Z_i$. Thus the $Z_i$ are irreducible components. Conversely, if $Y$ is an irreducible component of $X$, $Y \subseteq Z_i$ for some $i$ and $Y = Z_i$ by the definition of irreducible component.
@@ -2072,7 +2072,7 @@ Let $\mathfrak{l}$ be any field.
         \item $R$ is Noetherian.
         \item Every homogeneous ideal of $R_{\bullet}$ is finitely generated.
         \item Every chain $I_0\subseteq I_1 \subseteq \ldots$ of homogeneous ideals terminates.
-        \item Every set $\fM \neq \emptyset$ of homogeneous ideals has a $\subseteq$-maximal element.
+        \item Every set $\mathfrak{M} \neq \emptyset$ of homogeneous ideals has a $\subseteq$-maximal element.
         \item $R_0$  is Noetherian and the ideal $R_+$ is finitely generated.
         \item $R_0$ is Noetherian and $R / R_0$ is of finite type.
     \end{enumerate}