On center graphs of finite associative rings

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 G C ( R ) and the strict center graph denoted by G C ( R ) ¯ . We present the properties of G C ( R ) and explore its implications on the nature of Z ( R ) .   Specifically, we demonstrate that if G C ( R ) is complete, then Z ( R ) is an ideal in R .   Furthermore, in the case where R is a unital ring, the completeness of G C ( 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 G C ¯ ( T 2 ( p ) ) , where T 2 ( p ) represents the ring of upper-triangular matrices with entries in the ring ℤ / p ℤ 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.