Aydin, NesetDemir, CagriCamci, Didem Karalarlioglu2025-01-272025-01-2720181225-17632234-3024https://doi.org/10.4134/CKMS.c170409https://hdl.handle.net/20.500.12428/28782Let 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.eninfo:eu-repo/semantics/closedAccessprime idealsemiprime idealprime ring and semiprime ringArticle3341083109610.4134/CKMS.c170409N/AWOS:0004490617000042-s2.0-85056526802Q3