On The Spectrum Of A Ring
Posted: Sun Nov 22, 2015 12:34 am
- Let \( A \) be a ring. Show that \( Spec(A) \) is not connected if and only if \( A \) is isomorphic to the product of two non-zero rings \( R \) and \( S \), if and only if \( A \) contains non-trivial idempotents.
- If \( A \) is an integral domain, then show that \( Spec(A) \) is irreducible.
- Show that \( Spec(A) \) is irreducible if and only if \( A \) has a unique minimal prime ideal, if and only if the nilradical of \( A \) is a prime ideal.
- Prove that the spectrum \( Spec(L) \) of a local ring \( L \) is connected.