I would use both ambition and achievement as uncountables: Ambition is a desire for personal achievement. Activity can also be used as an uncountable but if there are various activities, the plural is fine as well. Student is incorrect without an article, which means that your last example is wrong.
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michaeltingDoes it just sound smoother or is it because its plural form is grammatically wrong?Ambitions is possible in other contexts: He is man of many ambitions.
michaeltingDo you have a specific reason for choosing something wiithout an article over something with an article?No. Speakers of English just have elected to use some words as uncountables and some others as countables. To make matters and life more interesting, some words can be used in both capacities. Language is not math with rules and reasons for every