- 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 \( \displaystyle \mathfrak{a} \).
- Show that \( \displaystyle r(\mathfrak{a}) \) is described as follows:
\[ \displaystyle r(\mathfrak{a}) = \left\{ x \in A \; \Big| \; \exists n \in \mathbb{N} : x^{n} \in \mathfrak{a} \right\} \] - What do we get in the special case \( \mathfrak{a} = \left( 0 \right) \)?
- Show that \( \displaystyle r(\mathfrak{a}) \) is described as follows:
- 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 \( \displaystyle A \).
Basic Ring Theory - 4 ( (Nil)Radical / Jacobson Radical )
-
- Community Team
- Posts: 314
- Joined: Tue Nov 10, 2015 8:25 pm
Basic Ring Theory - 4 ( (Nil)Radical / Jacobson Radical )
We follow the conventions of the previous posts regarding the ring \( \displaystyle A \).
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 9 guests