EXTENSIONS OF Σ-ZIP RINGS

Year 2017, , 1 - 22, 17.01.2017

Ouyang Lunqun Zhou Qiong Wu Jinfang

https://doi.org/10.24330/ieja.295657

Abstract

t. In this note we consider a new concept, so called Σ-zip ring, which
unifies zip rings and weak zip rings. We observe the basic properties of Σ-zip
rings, constructing typical examples. We study the relationship between the
Σ-zip property of a ring R and that of its Ore extensions and skew generalized
power series extensions. As a consequence, we obtain a generalization of several
known results relating to zip rings and weak zip rings. 

Keywords

Σ-zip ring, Ore extension

References

• [5] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta
• Math. Hungar., 107(3) (2005), 207-224.
• [6] C. Y. Hong, N. K. Kim, T. K. Kwak and Y. Lee, Extensions of zip rings, J.
• Pure Appl. Algebra, 195(3) (2005), 231-242.
• [7] Z. K. Liu, Triangular matrix representations of rings of generalized power series,
• Acta Math. Sin. (Engl. Ser.), 22(4) (2006), 989-998.
• [8] G. Marks, On 2-primal Ore extensions, Comm. Algebra, 29(5) (2001), 2113-
• 2123.
• [9] R. Mazurek and M. Ziembowski, Uniserial rings of skew generalized power
• series, J. Algebra, 318(2) (2007), 737-764
• [10] L. Ouyang, Ore extensions of weak zip rings, Glasg. Math. J., 51(3) (2009),
• 525-537.
• [11] M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser.
• A Math. Sci., 73(1) (1997), 14-17.
• [12] P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh.
• Math. Sem. Univ. Hamburg, 61 (1991), 15-33.
• [13] P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl.
• Algebra, 79(3) (1992), 293-312.
• [14] J. M. Zelmanowitz, The finite intersection property on annihilator right ideals,
• Proc. Amer. Math. Soc., 57(2) (1976), 213-216.