Remark 61.5.4. Let $A$ be a ring. Let $\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of $A_ w$ (Lemma 61.5.3) is at most $\kappa$. Namely, each $A_ E$ has cardinality at most $\kappa$ and the set of finite subsets of $A$ has cardinality at most $\kappa$ as well. Thus the result follows as $\kappa \otimes \kappa = \kappa$, see Sets, Section 3.6.

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