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Full-Text Articles in Physical Sciences and Mathematics
Upper Dimension And Bases Of Zero-Divisor Graphs Of Commutative Rings, S. Pirzada, M. Aijaza, Shane Redmond
Upper Dimension And Bases Of Zero-Divisor Graphs Of Commutative Rings, S. Pirzada, M. Aijaza, Shane Redmond
EKU Faculty and Staff Scholarship
For a commutative ring R with non-zero zero divisor set Z∗(R), the zero divisor graph of R is Γ(R) with vertex set Z∗(R), where two distinct vertices x and y are adjacent if and only if x y = 0. The upper dimension and the resolving number of a zero divisor graph Γ(R) of some rings are determined. We provide certain classes of rings which have the same upper dimension and metric dimension and give an example of a ring for which these values do not coincide. Further, we obtain some bounds for the upper dimension in zero divisor graphs …
Noncommutative Reality-Based Algebras Of Rank 6, Allen Herman, Mikhael Muzychuk, Bangteng Xu
Noncommutative Reality-Based Algebras Of Rank 6, Allen Herman, Mikhael Muzychuk, Bangteng Xu
EKU Faculty and Staff Scholarship
We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. …
The Recognition Problem For Table Algebras And Reality-Based Algebras, Allen Herman, Mikhail Muzychuk, Bangteng Xu
The Recognition Problem For Table Algebras And Reality-Based Algebras, Allen Herman, Mikhail Muzychuk, Bangteng Xu
EKU Faculty and Staff Scholarship
Given a finite-dimensional noncommutative semisimple algebra A over C with involution, we show that A always has a basis B for which ( A , B ) is a reality-based algebra. For algebras that have a one-dimensional representation δ , we show that there always exists an RBA-basis for which δ is a positive degree map. We characterize all RBA-bases of the 5-dimensional noncommutative semisimple algebra for which the algebra has a positive degree map, and give examples of RBA-bases of C ⊕ M n ( C ) for which the RBA has a positive degree map, for all n …