On essential extensions of reduced rings and domains.
A ring is a left essential extension of a reduced ring (domain) if it contains a left ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided ideal of the ring. It is known that if \(I\) is a reduced ring which is an essential ideal of a ring \(R\), then \(R\) itself is a reduced ring. The corresponding property for left essential extensions is known not to be true. This raises the question to describe the left essential extensions of reduced rings and domains which is addressed in this paper. The main result shows that any left essential extension of a reduced ring is a subdirect sum of rings which are left essential extensions of domains. This is achieved by firstly describing the relationship between minimal prime ideals of reduced rings and their left essential extensions. It is also shown that not all subdirect sums of rings which are left essential extensions of domains are left essential extensions of reduced rings. As an application of their results, it is shown that the upper radical class determined by the class of left essential extensions of domains is a special radical which is right but not left strong. Examples of such radical classes are known to exist, but they are relatively scarce.
scientific article; zbMATH DE number 2130532