# SOME EXAMPLES OF QUASI-ARMENDARIZ RINGS

• Hashemi, Ebrahim
• Published : 2007.08.31
• 64 6

#### Abstract

In [12], McCoy proved that if R is a commutative ring, then whenever g(x) is a zero-divisor in R[x], there exists a nonzero c $\in$ R such that cg(x) = 0. In this paper, first we extend this result to monoid rings. Then for a monoid M, we give some examples of M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings. Every reduced ring is M-quasi-Armendariz for any unique product monoid M and any strictly totally ordered monoid $(M,\;{\leq})$. Also $T_4(R)$ is M-quasi-Armendariz when R is reduced and M-Armendariz.

#### Keywords

Armendariz rings;quasi-Armendariz rings;monoid rings;unique product monoid rings

#### References

1. E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974), 470-473 https://doi.org/10.1017/S1446788700029190
2. D. D. Anderson and S. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 73 (1997), 14-17
3. G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra 265 (2003), no. 2, 103-122
4. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52 https://doi.org/10.1016/S0022-4049(01)00053-6
5. Y. Hirano, On ordered monoid rings over a quasi-Baer ring, Comm. Algebra 29 (2001), no. 5, 2089-2095 https://doi.org/10.1081/AGB-100002171
6. C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), no. 1, 103-122 https://doi.org/10.1081/AGB-120016752
7. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761 https://doi.org/10.1081/AGB-120013179
8. N. H Kim and Y. Lee, Armendariz rings and reduced rings. J. Algebra 223 (2000), no. 2, 477-488 https://doi.org/10.1006/jabr.1999.8017
9. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300
10. T. K. Lee and T. L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), no. 3, 583-593
11. N. H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly 49 (1942), 286-296 https://doi.org/10.2307/2303094
12. J. Okninski, Semigroup Algebra, Monographs and Textbooks in Pure and Applied Mathematics, 138. Marcel Dekker, Inc., New York, 1991
13. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17
14. P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl. Algebra 79 (1992), no. 3, 293-312 https://doi.org/10.1016/0022-4049(92)90056-L
15. L. Zhongkui, Armendariz rings relative to a monoid, Comm. Algebra 33 (2005), no. 3, 649-661 https://doi.org/10.1081/AGB-200049869
16. G. F Birkenmeier, J. Y. Kim, and J. K. Park, Polynomial extensions of Baer and quasi- Baer rings, J. Pure Appl. Algebra 159 (2001), no. 1, 25-42 https://doi.org/10.1016/S0022-4049(00)00055-4
17. D. S. Passman, The Algebraic structure of group rings, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977