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$A$. if $nilrad A = 0 ~\Longrightarrow$ $A$ has no nonzero nilpotents. @@ -157,11 +157,11 @@ $$nilrad A = \bigcap_{P \in Spec(A)} P$$ \end{defn} -\begin{defn}{idempotent} +\begin{defn}{}{idempotent} $e \in A$ such that $e^2=e$. \end{defn} -\begin{defn}{radical of an ideal} +\begin{defn}{}{radical of an ideal} $$rad I = \{ f \in A | f^n \in I~ \text{for some} n \}$$ $rad I$ is an ideal. @@ -171,7 +171,7 @@ $rad I = \bigcap_{\substack{P \in \operatorname{Spec}(A)\\ P \supset I}} P$ \end{defn} -\begin{defn}{local ring} +\begin{defn}{}{local ring} A \emph{local ring} has a unique maximal ideal. Notation: local ring $A$, its maximal ideal $\mM$, residue field $K=A/\mM$: @@ -233,7 +233,7 @@ -\subsection{Lemmas, propositions and corollaries} +\subsection{Zorn's lemma and Jacobson radicals} Let $\Sigma$ be a partially orddered set. Given subset $S \subset \Sigma$, an \emph{upper bound} of $S$ is an element $u \in \Sigma$ such that $s