Weakly clean and related rings / Qua Kiat Tat
Let R be an associative ring with identity. Let Id(R) and U(R) denote the set of idempotents and the set of units in R, respectively. An element x 2 R is said to be weakly clean if x can be written in the form x = u+e or x = u−e for some u 2 U(R) and e 2 Id(R). If x is represented uniquely in thi...
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| Format: | Thesis |
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2015
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| Online Access: | http://studentsrepo.um.edu.my/6548/ http://studentsrepo.um.edu.my/6548/1/weakly_clean_and_related_rings%2DQua_Kiat_Tat.pdf |
| _version_ | 1848773195606261760 |
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| author | Qua, Kiat Tat |
| author_facet | Qua, Kiat Tat |
| author_sort | Qua, Kiat Tat |
| building | UM Research Repository |
| collection | Online Access |
| description | Let R be an associative ring with identity. Let Id(R) and U(R) denote the
set of idempotents and the set of units in R, respectively. An element x 2 R is
said to be weakly clean if x can be written in the form x = u+e or x = u−e for
some u 2 U(R) and e 2 Id(R). If x is represented uniquely in this form, whether
x = u + e or x = u − e, then x is said to be uniquely weakly clean. We say that
x 2 R is pseudo weakly clean if x can be written in the form x = u+e+(1−e)rx
or x = u − e + (1 − e)rx for some u 2 U(R), e 2 Id(R) and r 2 R. For any
positive integer n, an element x 2 R is n-weakly clean if x = u1 +· · ·+un +e or
x = u1 + · · · + un − e for some u1, . . . ,un 2 U(R) and e 2 Id(R). The ring R is
said to be weakly clean (uniquely weakly clean, pseudo weakly clean, n-weakly
clean) if all of its elements are weakly clean (uniquely weakly clean, pseudo
weakly clean, n-weakly clean). Let g(x) be a polynomial in Z(R)[x] where Z(R)
denotes the centre of R. An element r 2 R is g(x)-clean if r = u + s for some
u 2 U(R) and s 2 R such that g(s) = 0 in R. The ring R is said to be g(x)-clean
if all of its elements are g(x)-clean. In this dissertation we investigate weakly
clean and related rings. We determine some characterisations and properties of
weakly clean, pseudo weakly clean, uniquely weakly clean, n-weakly clean and
g(x)-clean rings for certain types of g(x) 2 Z(R)[x]. Some generalisations of
results on clean and related rings are also obtained. |
| first_indexed | 2025-11-14T13:38:33Z |
| format | Thesis |
| id | um-6548 |
| institution | University Malaya |
| institution_category | Local University |
| last_indexed | 2025-11-14T13:38:33Z |
| publishDate | 2015 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | um-65482016-10-12T05:24:49Z Weakly clean and related rings / Qua Kiat Tat Qua, Kiat Tat Q Science (General) Let R be an associative ring with identity. Let Id(R) and U(R) denote the set of idempotents and the set of units in R, respectively. An element x 2 R is said to be weakly clean if x can be written in the form x = u+e or x = u−e for some u 2 U(R) and e 2 Id(R). If x is represented uniquely in this form, whether x = u + e or x = u − e, then x is said to be uniquely weakly clean. We say that x 2 R is pseudo weakly clean if x can be written in the form x = u+e+(1−e)rx or x = u − e + (1 − e)rx for some u 2 U(R), e 2 Id(R) and r 2 R. For any positive integer n, an element x 2 R is n-weakly clean if x = u1 +· · ·+un +e or x = u1 + · · · + un − e for some u1, . . . ,un 2 U(R) and e 2 Id(R). The ring R is said to be weakly clean (uniquely weakly clean, pseudo weakly clean, n-weakly clean) if all of its elements are weakly clean (uniquely weakly clean, pseudo weakly clean, n-weakly clean). Let g(x) be a polynomial in Z(R)[x] where Z(R) denotes the centre of R. An element r 2 R is g(x)-clean if r = u + s for some u 2 U(R) and s 2 R such that g(s) = 0 in R. The ring R is said to be g(x)-clean if all of its elements are g(x)-clean. In this dissertation we investigate weakly clean and related rings. We determine some characterisations and properties of weakly clean, pseudo weakly clean, uniquely weakly clean, n-weakly clean and g(x)-clean rings for certain types of g(x) 2 Z(R)[x]. Some generalisations of results on clean and related rings are also obtained. 2015 Thesis NonPeerReviewed application/pdf http://studentsrepo.um.edu.my/6548/1/weakly_clean_and_related_rings%2DQua_Kiat_Tat.pdf Qua, Kiat Tat (2015) Weakly clean and related rings / Qua Kiat Tat. PhD thesis, University of Malaya. http://studentsrepo.um.edu.my/6548/ |
| spellingShingle | Q Science (General) Qua, Kiat Tat Weakly clean and related rings / Qua Kiat Tat |
| title | Weakly clean and related rings / Qua Kiat Tat |
| title_full | Weakly clean and related rings / Qua Kiat Tat |
| title_fullStr | Weakly clean and related rings / Qua Kiat Tat |
| title_full_unstemmed | Weakly clean and related rings / Qua Kiat Tat |
| title_short | Weakly clean and related rings / Qua Kiat Tat |
| title_sort | weakly clean and related rings / qua kiat tat |
| topic | Q Science (General) |
| url | http://studentsrepo.um.edu.my/6548/ http://studentsrepo.um.edu.my/6548/1/weakly_clean_and_related_rings%2DQua_Kiat_Tat.pdf |