Generalisation of n-centraliser rings and their graphs

Chan, Tai Chong (2024) Generalisation of n-centraliser rings and their graphs. Master dissertation/thesis, UTAR.

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Abstract

Let Cent(R) denote the set of all distinct centralisers in a ring R. A ring R is said to be an n-centraliser ring if |Cent(R)| = n, where n 2 N. The question of how the number of distinct centralisers in a ring can influence its structure and commutativity has recently captured the attention of several researchers. Therefore, the study of the n-centraliser rings is a prospective research topic in ring theory. In this dissertation, we first investigate the characterisation for all n-centraliser finite rings for n 2 {6, 7, 8, 9, 10, 11} and compute their commuting probabilities. Subsequently, we classify the structures for all finite rings in which the cardinality of the maximal non-commuting set is 5. To extend the study of n-centraliser rings, we introduce the notion of (m, n)-centraliser rings, which is a generalisation of n-centraliser rings. For any m distinct elements r1, r2, ··· , rm in a ring R, the m-element centraliser of {r1, r2, ··· , rm} in R, denoted by CR({r1, r2, ··· , rm}), is defined as CR({r1, r2, ··· , rm}) = {s 2 R | sr1 = r1s, sr2 = r2s, ··· , srm = rms}, where m 2 N with m > 2. We denote the set of all distinct m-element centralisers in a ring R by m- Cent(R), where M E N with M> 2. A ring R is called an (m,n)- centraliser ring if |m - Cent(R)|=n, where n e N. Throughout this dissertation, we study the characterisation for some (m,n)- centraliser finite rings for n < 10. To establish an association between a graph and a ring, we introduce the idea of the non-centraliser graph of rings. The non-centraliser graph of a ring R, denoted by ⌥R, is a graph where the vertex set is R, and the edge set consists of {x, y}, where x, y are two distinct elements in R such that CR(x) 6= CR(y). In this dissertation, we discuss various graph theoretical properties of the non�centraliser graph of finite rings. Keywords: Finite ring, n-centraliser ring, (m, n)-centraliser ring, non-centraliser graph of ring, non-commuting set, commuting probability.

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