THE SOURCE OF SEMIPRIMENESS OF RINGS

Let R be an associative ring. We define a subset S-R of R as S-R = {a is an element of R vertical bar aRa = (0)} and call it the source of semiprimeness of R. We first examine some basic properties of the subset S-R in any ring R, and then define the notions such as R being a vertical bar S-R vertical bar-reduced ring, a vertical bar S-R vertical bar-domain and a vertical bar S-R vertical bar-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite vertical bar S-R vertical bar-domain is necessarily unitary, and is in fact a vertical bar S-R vertical bar-division ring. However, we provide an example showing that a finite vertical bar S-R vertical bar-division ring does not need to be commutative. All possible values for characteristics of unitary vertical bar S-R vertical bar-reduced rings and vertical bar S-R vertical bar-domains are also determined.