On Extensions of Right Symmetric Rings without Identity

Affiliation(s)

^{1}
Department of Mathematics, Umm Al-Qura University, Makkah, KSA.

^{2}
Department of Mathematics, King Abdulaziz University, Jeddah, KSA.

ABSTRACT

Let us call a ring*R *(without identity) to
be right symmetric if for any triple *a*,*b*,*c*,∈*R** abc *= 0* *then* acb* = 0. Such rings are neither
symmetric nor reversible (in general) but are semicommutative.
With an idempotent they take care of the sheaf representation as obtained by
Lambek. Klein 4-rings and their several generalizations and extensions are
proved to be members of such class of rings. An extension obtained is a McCoy
ring and its power series ring is also proved to be a McCoy ring.

Let us call a ring

Cite this paper

Shafee, B. and Nauman, S. (2014) On Extensions of Right Symmetric Rings without Identity.*Advances in Pure Mathematics*, **4**, 665-673. doi: 10.4236/apm.2014.412075.

Shafee, B. and Nauman, S. (2014) On Extensions of Right Symmetric Rings without Identity.

References

[1] Lambek, J. (1971) On the Representation of Modules by Sheaves of Factor Modules. Canadian Mathematical Bulletin, 14, 359-368.

http://dx.doi.org/10.4153/CMB-1971-065-1

[2] Ben Yakoub, L. and Louzari, M. (2009) On Extensions of Extended Symmetric and Reversible Rings. International Journal of Algebra, 3, 423-433.

[3] Huh, C., Kim, H.K., Kim, N.K. and Lee, Y. (2005) Basic Examples and Extensions of Symmetric Rings. Journal of Pure and Applied Algebra, 202, 154-167.

http://dx.doi.org/10.1016/j.jpaa.2005.01.009

[4] Kafkas, G., Ungor, B., Halicioglu, S. and Harmanici, A. (2011) Generalized Symmetric Rings. Algebra and Discrete mathematics, 12, 72-84.

[5] Kwak, T.K. (2007) Extensions of Extended Symmetric Rings. Bulletin of The Korean Mathematical Society, 4, 777-788.

http://dx.doi.org/10.4134/BKMS.2007.44.4.777

[6] Cohn, P.M. (1999) Reversible Rings. Bulletin of the London Mathematical Society, 31, 641-648.

http://dx.doi.org/10.1112/S0024609399006116

[7] Anderson, D. and Camillo, V. (1999) Semigroups and Rings Whose Zero Product Commutes. Communications in Algebra, 27, 2847-2852.

[8] Marks, G. (2002) Reversible and Symmetric Rings. Journal of Pure and Applied Algebra, 174, 311-318.

http://dx.doi.org/10.1016/S0022-4049(02)00070-1

[9] Bell, H.E. (1970) Near-rings, in Which Every Element Is a power of Itself. Bulletin of the Australian Mathematical Society, 2, 363-368.

http://dx.doi.org/10.1017/S0004972700042052

[10] Marks, G. (2003) A Taxonomy of 2-Primal Rings. Journal of Algebra, 266, 494-520.

http://dx.doi.org/10.1016/S0021-8693(03)00301-6

[11] Fakieh, W.M. and Nauman, S.K. (2013) Reversible Rings with Involutions and Some Minimalities. The Scientific World Journal, 2013, 8 pages, Article ID: 650702.

http://dx.doi.org/10.1155/2013/650702

[12] Rege, M.B. and Chhawchharia, S. (1997) Armendariz Rings. Proceedings of the Japan Academy, Series. A, Mathematical Sciences, 73, 14-17.

[13] Nielsen, P.P. (2006) Semi-Commutativity and McCoy Conditions. Journal of Algebra, 298, 134-141.

http://dx.doi.org/10.1016/j.jalgebra.2005.10.008

[14] Camillo, V. and Nielsen, P.P. (2008) McCoy Rings and Zero-Divisors. Journal of Pure and Applied Algebra, 212, 599-615.

http://dx.doi.org/10.1016/j.jpaa.2007.06.010

[1] Lambek, J. (1971) On the Representation of Modules by Sheaves of Factor Modules. Canadian Mathematical Bulletin, 14, 359-368.

http://dx.doi.org/10.4153/CMB-1971-065-1

[2] Ben Yakoub, L. and Louzari, M. (2009) On Extensions of Extended Symmetric and Reversible Rings. International Journal of Algebra, 3, 423-433.

[3] Huh, C., Kim, H.K., Kim, N.K. and Lee, Y. (2005) Basic Examples and Extensions of Symmetric Rings. Journal of Pure and Applied Algebra, 202, 154-167.

http://dx.doi.org/10.1016/j.jpaa.2005.01.009

[4] Kafkas, G., Ungor, B., Halicioglu, S. and Harmanici, A. (2011) Generalized Symmetric Rings. Algebra and Discrete mathematics, 12, 72-84.

[5] Kwak, T.K. (2007) Extensions of Extended Symmetric Rings. Bulletin of The Korean Mathematical Society, 4, 777-788.

http://dx.doi.org/10.4134/BKMS.2007.44.4.777

[6] Cohn, P.M. (1999) Reversible Rings. Bulletin of the London Mathematical Society, 31, 641-648.

http://dx.doi.org/10.1112/S0024609399006116

[7] Anderson, D. and Camillo, V. (1999) Semigroups and Rings Whose Zero Product Commutes. Communications in Algebra, 27, 2847-2852.

[8] Marks, G. (2002) Reversible and Symmetric Rings. Journal of Pure and Applied Algebra, 174, 311-318.

http://dx.doi.org/10.1016/S0022-4049(02)00070-1

[9] Bell, H.E. (1970) Near-rings, in Which Every Element Is a power of Itself. Bulletin of the Australian Mathematical Society, 2, 363-368.

http://dx.doi.org/10.1017/S0004972700042052

[10] Marks, G. (2003) A Taxonomy of 2-Primal Rings. Journal of Algebra, 266, 494-520.

http://dx.doi.org/10.1016/S0021-8693(03)00301-6

[11] Fakieh, W.M. and Nauman, S.K. (2013) Reversible Rings with Involutions and Some Minimalities. The Scientific World Journal, 2013, 8 pages, Article ID: 650702.

http://dx.doi.org/10.1155/2013/650702

[12] Rege, M.B. and Chhawchharia, S. (1997) Armendariz Rings. Proceedings of the Japan Academy, Series. A, Mathematical Sciences, 73, 14-17.

[13] Nielsen, P.P. (2006) Semi-Commutativity and McCoy Conditions. Journal of Algebra, 298, 134-141.

http://dx.doi.org/10.1016/j.jalgebra.2005.10.008

[14] Camillo, V. and Nielsen, P.P. (2008) McCoy Rings and Zero-Divisors. Journal of Pure and Applied Algebra, 212, 599-615.

http://dx.doi.org/10.1016/j.jpaa.2007.06.010