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 The Ohm-Rush content function III. Completion, globalization, and power-content algebras J. Korean Math. Soc.Published online August 12, 2021 Neil Epstein and Jay Shapiro George Mason University; George Mason University Abstract : One says that a ring homomorphism $R \rightarrow S$ is Ohm-Rush if extension commutes with arbitrary intersection of ideals, or equivalently if for any element $f\in S$, there is a unique smallest ideal of $R$ whose extension to $S$ contains $f$, called the content of $f$. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically yes' in dimension one, but no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property. Keywords : commutative algebra, content algebras, Ohm-Rush, completion, extended ideals, faithfully flat, Dedekind domain MSC numbers : 13B02 Full-Text :