Basic Ring Theory - 10

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Basic Ring Theory - 10

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Post by Tsakanikas Nickos »

Let \( B \) be a \( \mathbb{Z} \)-graded ring. Show that:
  1. If \( \mathfrak{a} \) is a homogeneous ideal of \( B \), then \( \sqrt{\mathfrak{a}} \) is also a homogeneous ideal of \( B \).
  2. Let \( \mathfrak{a} \) be a homogeneous ideal of \( B \). Then \( \mathfrak{a} \) is prime if and only if for all homogeneous elements \( a,b \in B \) the following implication holds: \( ab \in \mathfrak{a} \implies a \in \mathfrak{a} \text{ or } b \in \mathfrak{a} \)
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