On center graphs of finite associative rings - Jorf, M.; Oukhtite, L. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 512.5 We consider a finite associative ring R, which may have or may not have a unit element. We also examine its center denoted by Z(R). Our main focus is on the introduction of two distinct graphs associated with R, namely, the center graph denoted by GC(R) and the strict center graph denoted by GC(R). We present the properties of GC(R) and explore its implications on the nature of Z(R). Specifically, we demonstrate that if GC(R) is complete, then Z(R) is an ideal in R. Furthermore, in the case where R is a unital ring, the completeness of GC(R) leads to the conclusion that R is a commutative ring. As a specific application of our results, we provide an explicit construction of the graph GC(T_2(p)), where T_2(p) represents the ring of upper-triangular matrices with entries in the ring Z/pZ and p is a prime integer. In our investigations of the center graph and strict center graph, we aim to shed light on the properties of finite associative rings and their centers, thus providing valuable insights and applications in the ring theory.