diff --git a/algebra.tex b/algebra.tex
index 673eb79dc..9074ac7dd 100644
--- a/algebra.tex
+++ b/algebra.tex
@@ -19704,7 +19704,7 @@ \section{Mittag-Leffler systems}
 
 \noindent
 The purpose of this section is to define Mittag-Leffler systems
-and why it is a useful property.
+and why this is a useful notion.
 
 \medskip\noindent
 In the following, $I$ will be a directed set, see
@@ -19714,9 +19714,9 @@ \section{Mittag-Leffler systems}
 Categories, Definition \ref{categories-definition-directed-system}.
 This is a directed inverse system as we assumed $I$ directed
 (Categories, Definition \ref{categories-definition-directed-system}).
-For each $i \in I$, the
-images $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$ form a decreasing
-family.  Let $A'_i = \bigcap_{j \geq i} \varphi_{ji}(A_j)$.
+For each $i \in I$, the images $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$
+form a decreasing directed family of subsets (or submodules) of $A_i$. Let
+$A'_i = \bigcap_{j \geq i} \varphi_{ji}(A_j)$.
 Then $\varphi_{ji}(A'_j) \subset A'_i$ for $j \geq i$, hence by restricting
 we get a directed inverse system $(A'_i, \varphi_{ji}|_{A'_j})$.
 From the construction of the limit of an inverse system in the category
@@ -19729,8 +19729,8 @@ \section{Mittag-Leffler systems}
 \label{definition-ML-system}
 Let $(A_i, \varphi_{ji})$ be a directed inverse system of sets over $I$.  Then
 we say  $(A_i, \varphi_{ji})$ is {\it Mittag-Leffler inverse system} if for
-each $i \in I$, the decreasing family $\varphi_{ji}(A_j) \subset A_i$ for $j
-\geq i$ stabilizes.  Explicitly, this means that for each $i \in I$, there
+each $i \in I$, the family $\varphi_{ji}(A_j) \subset A_i$ for
+$j \geq i$ stabilizes.  Explicitly, this means that for each $i \in I$, there
 exists $j \geq i$ such that for $k \geq j$ we have $\varphi_{ki}(A_k) =
 \varphi_{ji}( A_j)$.  If $(A_i, \varphi_{ji})$ is a directed inverse system
 of modules over a ring $R$, we say that it is Mittag-Leffler if the underlying