Basic Ring Theory - 5
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Basic Ring Theory - 5
Let \( A \) be a ring. Suppose that for every prime ideal \( \mathfrak{p} \) of \( A \) the local ring \( A_{\mathfrak{p}} \) has no non-zero nilpotent elements. Prove that then \( A \) itself has no non-zero nilpotent elements.
If for every prime ideal \( \mathfrak{p} \) of \( A \) the local ring \( A_{\mathfrak{p}} \) is an integral domain, is \( A \) itself necessarily an integral domain?
If for every prime ideal \( \mathfrak{p} \) of \( A \) the local ring \( A_{\mathfrak{p}} \) is an integral domain, is \( A \) itself necessarily an integral domain?
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