Prüfer ring and Arithmetical ring

Abstract

In this paper, it was presented a study of Prüfer ring and what conditions should be applied for the quotient ring to be an Arithmetical ring through the theorem (1.3), as a result of this theorem if  is Prüfer domain, then the quotient ring is an Arithmetical ring.

New descriptions of Arithmetical ring in case it was reduced have also developed through the theorem (4.3) and result (7.3) and a Noetherian ring is a finite direct product of Noetherian rings each of them contains a unique minimal prime ideal within certain conditions through the theorem (8.3), it is a generalization that an Artin ring is a finite direct product of Artin local rings and then an Arithmetical Noetherian ring is a finite direct product of Arithmetical Noetherian rings each of them contains a unique minimal prime ideal as in the result (11.3).

Finding a criterion helps us to test a ring is not an Arithmetical ring as the result (5.3).