Research Article

### CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP

Year 2017, Volume 21, Issue 21, 121 - 126, 17.01.2017

### Abstract

A reality-based algebra (RBA) is a finite-dimensional associative algebra that has a distinguished basis B containing 1A, where 1A is the identity element of A, that is closed under a pseudo-inverse condition. If the RBA has a one-dimensional representation taking positive values on B, then we say that the RBA has a positive degree map. When the structure constants relative to a standardized basis of an RBA with positive degree map are all integers, we say that the RBA is integral. Group algebras of finite groups are examples of integral RBAs with a positive degree map, and so it is natural to ask if properties known to hold for group algebras also hold for integral RBAs with positive degree map. In this article we show that every central torsion unit of an integral RBA with algebraic integer coefficients is a trivial unit of the form ζb, for some ζ is a root of unit in C and b is an element of degree 1 in B.

### References

• [1] H. I. Blau, Table algebras, European J. Combin., 30(6) (2009), 1426-1455.
• [2] A. Herman and G. Singh, On the torsion units of integral adjacency algebras of
• finite association schemes, Algebra, 2014 (2014), Article ID 842378, 5 pages.
• [3] D. G. Higman, Coherent algebras, Linear Algebra Appl., 93 (1987), 209-239.
• [4] G. Singh, Torsion Units of Integral Group Rings and Scheme Rings, Ph.D.
• Thesis, University of Regina, 2015.
• [5] M. Takesaki, Theory of Operator Algebras I, Encyclopaedia of Mathematical
• Sciences, 124, Springer-Verlag, Berlin, 1979.
• [6] B. Xu, On isomorphisms between integral table algebras and applications to
• finite groups and association schemes, Comm. Algebra, 42(12) (2014), 5249-5263.
Year 2017, Volume 21, Issue 21, 121 - 126, 17.01.2017

### References

• [1] H. I. Blau, Table algebras, European J. Combin., 30(6) (2009), 1426-1455.
• [2] A. Herman and G. Singh, On the torsion units of integral adjacency algebras of
• finite association schemes, Algebra, 2014 (2014), Article ID 842378, 5 pages.
• [3] D. G. Higman, Coherent algebras, Linear Algebra Appl., 93 (1987), 209-239.
• [4] G. Singh, Torsion Units of Integral Group Rings and Scheme Rings, Ph.D.
• Thesis, University of Regina, 2015.
• [5] M. Takesaki, Theory of Operator Algebras I, Encyclopaedia of Mathematical
• Sciences, 124, Springer-Verlag, Berlin, 1979.
• [6] B. Xu, On isomorphisms between integral table algebras and applications to
• finite groups and association schemes, Comm. Algebra, 42(12) (2014), 5249-5263.

### Details

Subjects Mathematics Articles Allen Herman This is me Gurmail Singh This is me January 17, 2017 Year 2017, Volume 21, Issue 21

### Cite

 Bibtex @research article { ieja296156, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2017}, volume = {21}, number = {21}, pages = {121 - 126}, doi = {10.24330/ieja.296156}, title = {CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP}, key = {cite}, author = {Herman, Allen and Singh, Gurmail} } APA Herman, A. & Singh, G. (2017). CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP . International Electronic Journal of Algebra , 21 (21) , 121-126 . DOI: 10.24330/ieja.296156 MLA Herman, A. , Singh, G. "CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP" . International Electronic Journal of Algebra 21 (2017 ): 121-126 Chicago Herman, A. , Singh, G. "CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP". International Electronic Journal of Algebra 21 (2017 ): 121-126 RIS TY - JOUR T1 - CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP AU - AllenHerman, GurmailSingh Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.296156 DO - 10.24330/ieja.296156 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 121 EP - 126 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.296156 UR - https://doi.org/10.24330/ieja.296156 Y2 - 2016 ER - EndNote %0 International Electronic Journal of Algebra CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP %A Allen Herman , Gurmail Singh %T CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.296156 %U 10.24330/ieja.296156 ISNAD Herman, Allen , Singh, Gurmail . "CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP". International Electronic Journal of Algebra 21 / 21 (January 2017): 121-126 . https://doi.org/10.24330/ieja.296156 AMA Herman A. , Singh G. CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. IEJA. 2017; 21(21): 121-126. Vancouver Herman A. , Singh G. CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP. International Electronic Journal of Algebra. 2017; 21(21): 121-126. IEEE A. Herman and G. Singh , "CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP", International Electronic Journal of Algebra, vol. 21, no. 21, pp. 121-126, Jan. 2017, doi:10.24330/ieja.296156