Prime one-sided ideals of a finite normalizing extension (Q1085256)

A right ideal I of a ring R is called prime if, for any two right ideals X and Y of R, XY\(\subseteq I\) implies that \(X\subseteq I\) or \(Y\subseteq I\). In this note, some well-known facts about prime two-sided ideals in finite normalizing (centralizing, intermediate) ring extensions, such as ''Lying Over'', ''Cutting Down'', etc., are extended to the case of prime right ideals, usually under additional assumptions on the ring extension in question.