A ruthenium-carrying alumina, which is prepared by suspending alumina in a solution containing trivalent ruthenium and adding a base to the suspension, is provided. This ruthenium-carrying alumina is useful as a catalyst for oxidizing alcohols by contacting the alcohols with molecular oxygen, and can be used for oxidizing the alcohols at a high conversion to produce ketones, aldehydes, carboxylic acids, etc. with good productivity.