Basic Ring Theory - 4 ( (Nil)Radical / Jacobson Radical )

[Tsakanikas Nickos]

We follow the conventions of the previous posts regarding the ring \( \displaystyle A \).

• The radical \( \displaystyle r( \mathfrak{a} \)) of an ideal \( \displaystyle \mathfrak{a} \) of \( \displaystyle A \) is defined as the intersection of all prime ideals of \( \displaystyle A \) containing \( \displaystyle \mathfrak{a} \).
  1. Show that \( \displaystyle r(\mathfrak{a}) \) is described as follows:
     \[ \displaystyle r(\mathfrak{a}) = \left\{ x \in A \; \Big| \; \exists n \in \mathbb{N} : x^{n} \in \mathfrak{a} \right\} \]
  2. What do we get in the special case \( \mathfrak{a} = \left( 0 \right) \)?
• The Jacobson radical \( \mathfrak{R} \) of \( \displaystyle A \) is defined as the intersection of all maximal ideals of \( \displaystyle A \). Show that \( \mathfrak{R} \) is described as follows:
  \[ \displaystyle \mathfrak{R} = \left\{ x \in A \; \Big| \; 1_{A} - xy \in U(A) \, , \, \forall y \in A \right\} \]where \( \displaystyle U(A) \) is the group of all invertible elements of \( \displaystyle A \).