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replaced http://mathoverflow.net/ with https://mathoverflow.net/
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Not necessarily. See Examples of common false beliefs in mathematicsExamples of common false beliefs in mathematics and the answer by JSE and the reply to his example by Simon Wadsley.

Not necessarily. See Examples of common false beliefs in mathematics and the answer by JSE and the reply to his example by Simon Wadsley.

Not necessarily. See Examples of common false beliefs in mathematics and the answer by JSE and the reply to his example by Simon Wadsley.

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Not necessarily. See Examples of common false beliefs in mathematics and the answer by JSE and the reply to his example by Simon Wadsley.