Ring Theory - Some Useful Theorems

Some Useful Theorems

General:

• Isomorphism theorems for rings
• Nakayama's lemma

Structure theorems:

• The Artin–Wedderburn theorem determines the structure of semisimple rings.
• The Jacobson density theorem determines the structure of primitive rings.
• Goldie's theorem determines the structure of semiprime Goldie rings.
• The Zariski-Samuel theorem determines the structure of a commutative principal ideal rings.
• The Hopkins–Levitzki theorem gives necessary and sufficient conditions for a Noetherian ring to be an Artinian ring.
• Morita theory consists of theorems determining when two rings have "equivalent" module categories.
• Wedderburn's little theorem states that finite domains are fields.