Simple, prime and semiprime rings

Abstract

This chapter discusses simple, prime, and semiprime rings. A ring R is called “simple” if it has no proper two-sided ideals and if R 2 ≠ 0. Any field and any skew field are simple rings. The ring of n × n matrices over a ring R is simple only if R is simple. A center of a simple ring is either a field or zero. In the former case, a unit of the center is a unit of the ring. In addition, any simple ring with a unit can be considered an algebra over a field (its center). If the center of a simple ring is zero, then it still has an algebra structure over a field—its centroid. To any ring property can be associated the map that associates to a ring the biggest ideal obeying as a ring this property. If such a map proves to be a radical, then the property is called a “radical property.”