The n-zero-divisor graph of a commutative semigroup
Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph Γ(S) with vertices Z(S) ∗ = Z(S) \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introd...
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2022
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| Online Access: | http://hdl.handle.net/11073/25070 |
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| _version_ | 1864513435860992000 |
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| author | Anderson, David F. |
| author2 | Badawi, Ayman |
| author2_role | author |
| author_facet | Anderson, David F. Badawi, Ayman |
| author_role | author |
| dc.creator.none.fl_str_mv | Anderson, David F. Badawi, Ayman |
| dc.date.none.fl_str_mv | 2022-11-28T11:33:15Z 2022-11-28T11:33:15Z 2022-04-16 |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | Anderson, D. F., & Badawi, A. (2022). The n-zero-divisor graph of a commutative semigroup. In Communications in Algebra (Vol. 50, Issue 10, pp. 4155–4177). Informa UK Limited. https://doi.org/10.1080/00927872.2022.2057521 1532-4125 http://hdl.handle.net/11073/25070 10.1080/00927872.2022.2057521 |
| dc.language.none.fl_str_mv | en_US |
| dc.publisher.none.fl_str_mv | Taylor and Francis |
| dc.relation.none.fl_str_mv | https://doi.org/10.1080/00927872.2022.2057521 |
| dc.subject.none.fl_str_mv | Idempotent elements Zero-divisors Commutative semigroup with zero Commutative ring with identity Von Neumann regular ring π-regular ring Zero-divisor graph Annihilator graph Extended zero-divisor graph Congruence-based zero-divisor graph |
| dc.title.none.fl_str_mv | The n-zero-divisor graph of a commutative semigroup |
| dc.type.none.fl_str_mv | Peer-Reviewed Postprint info:eu-repo/semantics/publishedVersion info:eu-repo/semantics/article |
| description | Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph Γ(S) with vertices Z(S) ∗ = Z(S) \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph Γn(S) with vertices Zn(S) ∗ = {x n | x ∈ Z(S)} \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. Thus each Γn(S) is an induced subgraph of Γ(S) = Γ1(S). We pay particular attention to diam(Γn(S)), gr(Γn(S)), and the case when S is a commutative ring with 1 6= 0. We also consider several other types of “n-zero-divisor” graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent. |
| format | article |
| id | aus_1e8c2f09439adea56af818d644543cea |
| identifier_str_mv | Anderson, D. F., & Badawi, A. (2022). The n-zero-divisor graph of a commutative semigroup. In Communications in Algebra (Vol. 50, Issue 10, pp. 4155–4177). Informa UK Limited. https://doi.org/10.1080/00927872.2022.2057521 1532-4125 10.1080/00927872.2022.2057521 |
| language_invalid_str_mv | en_US |
| network_acronym_str | aus |
| network_name_str | aus |
| oai_identifier_str | oai:repository.aus.edu:11073/25070 |
| publishDate | 2022 |
| publisher.none.fl_str_mv | Taylor and Francis |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| spelling | The n-zero-divisor graph of a commutative semigroupAnderson, David F.Badawi, AymanIdempotent elementsZero-divisorsCommutative semigroup with zeroCommutative ring with identityVon Neumann regular ringπ-regular ringZero-divisor graphAnnihilator graphExtended zero-divisor graphCongruence-based zero-divisor graphLet S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph Γ(S) with vertices Z(S) ∗ = Z(S) \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph Γn(S) with vertices Zn(S) ∗ = {x n | x ∈ Z(S)} \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. Thus each Γn(S) is an induced subgraph of Γ(S) = Γ1(S). We pay particular attention to diam(Γn(S)), gr(Γn(S)), and the case when S is a commutative ring with 1 6= 0. We also consider several other types of “n-zero-divisor” graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent.Taylor and Francis2022-11-28T11:33:15Z2022-11-28T11:33:15Z2022-04-16Peer-ReviewedPostprintinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfAnderson, D. F., & Badawi, A. (2022). The n-zero-divisor graph of a commutative semigroup. In Communications in Algebra (Vol. 50, Issue 10, pp. 4155–4177). Informa UK Limited. https://doi.org/10.1080/00927872.2022.20575211532-4125http://hdl.handle.net/11073/2507010.1080/00927872.2022.2057521en_UShttps://doi.org/10.1080/00927872.2022.2057521oai:repository.aus.edu:11073/250702024-08-22T12:01:36Z |
| spellingShingle | The n-zero-divisor graph of a commutative semigroup Anderson, David F. Idempotent elements Zero-divisors Commutative semigroup with zero Commutative ring with identity Von Neumann regular ring π-regular ring Zero-divisor graph Annihilator graph Extended zero-divisor graph Congruence-based zero-divisor graph |
| status_str | publishedVersion |
| title | The n-zero-divisor graph of a commutative semigroup |
| title_full | The n-zero-divisor graph of a commutative semigroup |
| title_fullStr | The n-zero-divisor graph of a commutative semigroup |
| title_full_unstemmed | The n-zero-divisor graph of a commutative semigroup |
| title_short | The n-zero-divisor graph of a commutative semigroup |
| title_sort | The n-zero-divisor graph of a commutative semigroup |
| topic | Idempotent elements Zero-divisors Commutative semigroup with zero Commutative ring with identity Von Neumann regular ring π-regular ring Zero-divisor graph Annihilator graph Extended zero-divisor graph Congruence-based zero-divisor graph |
| url | http://hdl.handle.net/11073/25070 |