H. Chimal-Dzul and S. Szabo,
Minimal reflexive nonsemicommutative rings,
J. Algebra Appl., 21(4) (2022), 2250084 (15pp).
P. M. Cohn,
Reversible rings,
Bull. London Math. Soc., 31(6) (1999), 641-648.
L. Creedon, K. Hughes and S. Szabo,
A comparison of group algebras of dihedral and quaternion groups,
Appl. Algebra Engrg. Comm. Comput., 32(3) (2021), 245-264.
J. M. Grau, C. Miguel and A. M. Oller-Marcen,
On the structure of quaternion rings over $\mathbb{Z}/n\mathbb{Z}$,
Adv. Appl. Clifford Algebr., 25(4) (2015), 875-887.
J. M. Grau, C. Miguel and A. M. Oller-Marcen,
On the zero divisor graphs of the ring of Lipschitz integers modulo $n$,
Adv. Appl. Clifford Algebr., 27(2) (2017), 1191-1202.
J. M. Grau, C. Miguel and A. M. Oller-Marcen,
Generalized quaternion rings over $\mathbb Z/n\mathbb Z$ for an odd $n$,
Adv. Appl. Clifford Algebr., 28(1) (2018), 17 (14pp).
S. U. Hwang, Y. C. Jeon and Y. Lee,
Structure and topological conditions of NI rings,
J. Algebra, 302(1) (2006), 186-199.
J. Lambek,
On the representation of modules by sheaves of factor modules,
Canad. Math. Bull., 14 (1971), 359-368.
G. Marks,
Reversible and symmetric rings,
J. Pure Appl. Algebra, 174(3) (2002), 311-318.
G. Marks,
A taxonomy of 2-primal rings,
J. Algebra, 266(2) (2003), 494-520.
G. Mason,
Reflexive ideals,
Comm. Algebra, 9(17) (1981), 1709-1724.
S. Szabo,
Minimal reversible nonsymmetric rings,
J. Pure Appl. Algebra, 223(11) (2019), 4583-4591.
S. Szabo,
Some minimal rings related to 2-primal rings,
Comm. Algebra, 47(3) (2019), 1287-1298.
J. Voight, Characterizing quaternion rings over an arbitrary base, J. Reine Angew. Math., 657 (2011), 113-134.
Minimal rings related to generalized quaternion rings
Year 2023,
Volume: 34 Issue: 34, 88 - 111, 10.07.2023
The family of rings of the form
\frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left \langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \right \rangle}
is investigated which contains the generalized Hamilton quaternions over $\Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $\Z_{p^s}$ are characterized.
H. Chimal-Dzul and S. Szabo,
Minimal reflexive nonsemicommutative rings,
J. Algebra Appl., 21(4) (2022), 2250084 (15pp).
P. M. Cohn,
Reversible rings,
Bull. London Math. Soc., 31(6) (1999), 641-648.
L. Creedon, K. Hughes and S. Szabo,
A comparison of group algebras of dihedral and quaternion groups,
Appl. Algebra Engrg. Comm. Comput., 32(3) (2021), 245-264.
J. M. Grau, C. Miguel and A. M. Oller-Marcen,
On the structure of quaternion rings over $\mathbb{Z}/n\mathbb{Z}$,
Adv. Appl. Clifford Algebr., 25(4) (2015), 875-887.
J. M. Grau, C. Miguel and A. M. Oller-Marcen,
On the zero divisor graphs of the ring of Lipschitz integers modulo $n$,
Adv. Appl. Clifford Algebr., 27(2) (2017), 1191-1202.
J. M. Grau, C. Miguel and A. M. Oller-Marcen,
Generalized quaternion rings over $\mathbb Z/n\mathbb Z$ for an odd $n$,
Adv. Appl. Clifford Algebr., 28(1) (2018), 17 (14pp).
S. U. Hwang, Y. C. Jeon and Y. Lee,
Structure and topological conditions of NI rings,
J. Algebra, 302(1) (2006), 186-199.
J. Lambek,
On the representation of modules by sheaves of factor modules,
Canad. Math. Bull., 14 (1971), 359-368.
G. Marks,
Reversible and symmetric rings,
J. Pure Appl. Algebra, 174(3) (2002), 311-318.
G. Marks,
A taxonomy of 2-primal rings,
J. Algebra, 266(2) (2003), 494-520.
G. Mason,
Reflexive ideals,
Comm. Algebra, 9(17) (1981), 1709-1724.
S. Szabo,
Minimal reversible nonsymmetric rings,
J. Pure Appl. Algebra, 223(11) (2019), 4583-4591.
S. Szabo,
Some minimal rings related to 2-primal rings,
Comm. Algebra, 47(3) (2019), 1287-1298.
J. Voight, Characterizing quaternion rings over an arbitrary base, J. Reine Angew. Math., 657 (2011), 113-134.
Grau, J. M., Oller-marcen, A. M., & Szabo, S. (2023). Minimal rings related to generalized quaternion rings. International Electronic Journal of Algebra, 34(34), 88-111. https://doi.org/10.24330/ieja.1281705
AMA
Grau JM, Oller-marcen AM, Szabo S. Minimal rings related to generalized quaternion rings. IEJA. July 2023;34(34):88-111. doi:10.24330/ieja.1281705
Chicago
Grau, Jose Maria, Antonio M. Oller-marcen, and Steve Szabo. “Minimal Rings Related to Generalized Quaternion Rings”. International Electronic Journal of Algebra 34, no. 34 (July 2023): 88-111. https://doi.org/10.24330/ieja.1281705.
EndNote
Grau JM, Oller-marcen AM, Szabo S (July 1, 2023) Minimal rings related to generalized quaternion rings. International Electronic Journal of Algebra 34 34 88–111.
IEEE
J. M. Grau, A. M. Oller-marcen, and S. Szabo, “Minimal rings related to generalized quaternion rings”, IEJA, vol. 34, no. 34, pp. 88–111, 2023, doi: 10.24330/ieja.1281705.
ISNAD
Grau, Jose Maria et al. “Minimal Rings Related to Generalized Quaternion Rings”. International Electronic Journal of Algebra 34/34 (July 2023), 88-111. https://doi.org/10.24330/ieja.1281705.
JAMA
Grau JM, Oller-marcen AM, Szabo S. Minimal rings related to generalized quaternion rings. IEJA. 2023;34:88–111.
MLA
Grau, Jose Maria et al. “Minimal Rings Related to Generalized Quaternion Rings”. International Electronic Journal of Algebra, vol. 34, no. 34, 2023, pp. 88-111, doi:10.24330/ieja.1281705.
Vancouver
Grau JM, Oller-marcen AM, Szabo S. Minimal rings related to generalized quaternion rings. IEJA. 2023;34(34):88-111.