Minimal prime ideals of reduced rings

ABSTRACT

Let R be a reduced ring with Q its Martindale symmetric ring of quotients, and let B be the complete Boolean algebra of all idempotents in C, where C is the extended centroid of R. It is proved that every minimal prime ideal of R must be of the form mQ∩R for some maximal ideal m of B but the converse is in general not true. In addition, if R is centrally closed or has only finitely many minimal prime ideals, then the converse also holds. By applying the explicit expression, many properties of minimal prime ideals of reduced rings are realized more easily.

KEYWORDS:

• Complete Boolean algebra
• extended centroid
• Martindale symmetric ring of quotients
• minimal prime ideal
• reduced ring
• skew derivation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U99
• 16N60

Acknowledgments

The authors are grateful to the referee for carefully reading their manuscript. The work was supported in part by the Ministry of Science and Technology of Taiwan (MOST 105-2115-M-002 -003 -MY2) and the National Center for Theoretical Sciences, Taipei Office.