Generalisation of n-centraliser rings and their graphs
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 seve...
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| Format: | Final Year Project / Dissertation / Thesis |
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2024
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| Online Access: | http://eprints.utar.edu.my/6339/ http://eprints.utar.edu.my/6339/1/6._Revised_Dissertation_(Chan_Tai_Chong)_(1).pdf |
| _version_ | 1848886651069136896 |
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| author | Chan, Tai Chong |
| author_facet | Chan, Tai Chong |
| author_sort | Chan, Tai Chong |
| building | UTAR Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-15T19:41:53Z |
| format | Final Year Project / Dissertation / Thesis |
| id | utar-6339 |
| institution | Universiti Tunku Abdul Rahman |
| institution_category | Local University |
| last_indexed | 2025-11-15T19:41:53Z |
| publishDate | 2024 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | utar-63392024-04-14T11:00:52Z Generalisation of n-centraliser rings and their graphs Chan, Tai Chong Q Science (General) QA Mathematics 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. 2024 Final Year Project / Dissertation / Thesis NonPeerReviewed application/pdf http://eprints.utar.edu.my/6339/1/6._Revised_Dissertation_(Chan_Tai_Chong)_(1).pdf Chan, Tai Chong (2024) Generalisation of n-centraliser rings and their graphs. Master dissertation/thesis, UTAR. http://eprints.utar.edu.my/6339/ |
| spellingShingle | Q Science (General) QA Mathematics Chan, Tai Chong Generalisation of n-centraliser rings and their graphs |
| title | Generalisation of n-centraliser rings and their graphs |
| title_full | Generalisation of n-centraliser rings and their graphs |
| title_fullStr | Generalisation of n-centraliser rings and their graphs |
| title_full_unstemmed | Generalisation of n-centraliser rings and their graphs |
| title_short | Generalisation of n-centraliser rings and their graphs |
| title_sort | generalisation of n-centraliser rings and their graphs |
| topic | Q Science (General) QA Mathematics |
| url | http://eprints.utar.edu.my/6339/ http://eprints.utar.edu.my/6339/1/6._Revised_Dissertation_(Chan_Tai_Chong)_(1).pdf |