12 research outputs found
A one-sided Prime Ideal Principle for noncommutative rings
Completely prime right ideals are introduced as a one-sided generalization of
the concept of a prime ideal in a commutative ring. Some of their basic
properties are investigated, pointing out both similarities and differences
between these right ideals and their commutative counterparts. We prove the
Completely Prime Ideal Principle, a theorem stating that right ideals that are
maximal in a specific sense must be completely prime. We offer a number of
applications of the Completely Prime Ideal Principle arising from many diverse
concepts in rings and modules. These applications show how completely prime
right ideals control the one-sided structure of a ring, and they recover
earlier theorems stating that certain noncommutative rings are domains (namely,
proper right PCI rings and rings with the right restricted minimum condition
that are not right artinian). In order to provide a deeper understanding of the
set of completely prime right ideals in a general ring, we study the special
subset of comonoform right ideals.Comment: 38 page