Zero-divisor graphs of reduced Rickart *-rings
For a ring A with an involution *, the zero-divisor graph of A, Γ*(A), is the graph whose vertices are the nonzero left zero-divisors in A such that distinct vertices x and y are adjacent if and only if xy* = 0. In this paper, we study the zero-divisor graph of a Rickart *-ring having no nonzero nil...
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| Format: | Article |
| Language: | English |
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University of Zielona Góra
2017-06-01
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| Series: | Discussiones Mathematicae - General Algebra and Applications |
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| Online Access: | https://doi.org/10.7151/dmgaa.1265 |
| _version_ | 1827849623308337152 |
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| author | Patil A.A. Waphare B.N. |
| author_facet | Patil A.A. Waphare B.N. |
| author_sort | Patil A.A. |
| collection | DOAJ |
| description | For a ring A with an involution *, the zero-divisor graph of A, Γ*(A), is the graph whose vertices are the nonzero left zero-divisors in A such that distinct vertices x and y are adjacent if and only if xy* = 0. In this paper, we study the zero-divisor graph of a Rickart *-ring having no nonzero nilpotent element. The distance, diameter, and cycles of Γ*(A) are characterized in terms of the collection of prime strict ideals of A. In fact, we prove that the clique number of Γ*(A) coincides with the cellularity of the hullkernel topological space Σ(A) of the set of prime strict ideals of A, where cellularity of the topological space is the smallest cardinal number m such that every family of pairwise disjoint non-empty open subsets of the space have cardinality at most m. |
| first_indexed | 2024-03-12T10:07:52Z |
| format | Article |
| id | doaj.art-d4e43c2b64c848f292a10f1bed9a048b |
| institution | Directory Open Access Journal |
| issn | 2084-0373 |
| language | English |
| last_indexed | 2024-03-12T10:07:52Z |
| publishDate | 2017-06-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae - General Algebra and Applications |
| spelling | doaj.art-d4e43c2b64c848f292a10f1bed9a048b2023-09-02T11:07:12ZengUniversity of Zielona GóraDiscussiones Mathematicae - General Algebra and Applications2084-03732017-06-01371314310.7151/dmgaa.1265dmgaa.1265Zero-divisor graphs of reduced Rickart *-ringsPatil A.A.0Waphare B.N.1Department of Mathematics, Garware College of Commerce, Pune-411004, IndiaCenter for Advanced Studies in Mathematics, Department of Mathematics, Savitribai Phule Pune University, Pune-411007, India, bnwaph@math.unipune.ac.inFor a ring A with an involution *, the zero-divisor graph of A, Γ*(A), is the graph whose vertices are the nonzero left zero-divisors in A such that distinct vertices x and y are adjacent if and only if xy* = 0. In this paper, we study the zero-divisor graph of a Rickart *-ring having no nonzero nilpotent element. The distance, diameter, and cycles of Γ*(A) are characterized in terms of the collection of prime strict ideals of A. In fact, we prove that the clique number of Γ*(A) coincides with the cellularity of the hullkernel topological space Σ(A) of the set of prime strict ideals of A, where cellularity of the topological space is the smallest cardinal number m such that every family of pairwise disjoint non-empty open subsets of the space have cardinality at most m.https://doi.org/10.7151/dmgaa.1265reduced ringrickart *-ringzero-divisor graphprime strict idealsprimary: 16w10secondary: 05c25, 05c15 |
| spellingShingle | Patil A.A. Waphare B.N. Zero-divisor graphs of reduced Rickart *-rings Discussiones Mathematicae - General Algebra and Applications reduced ring rickart *-ring zero-divisor graph prime strict ideals primary: 16w10 secondary: 05c25, 05c15 |
| title | Zero-divisor graphs of reduced Rickart *-rings |
| title_full | Zero-divisor graphs of reduced Rickart *-rings |
| title_fullStr | Zero-divisor graphs of reduced Rickart *-rings |
| title_full_unstemmed | Zero-divisor graphs of reduced Rickart *-rings |
| title_short | Zero-divisor graphs of reduced Rickart *-rings |
| title_sort | zero divisor graphs of reduced rickart rings |
| topic | reduced ring rickart *-ring zero-divisor graph prime strict ideals primary: 16w10 secondary: 05c25, 05c15 |
| url | https://doi.org/10.7151/dmgaa.1265 |
| work_keys_str_mv | AT patilaa zerodivisorgraphsofreducedrickartrings AT wapharebn zerodivisorgraphsofreducedrickartrings |