Browse > Article
http://dx.doi.org/10.4134/JKMS.2016.53.2.381

ON ANNIHILATIONS OF IDEALS IN SKEW MONOID RINGS  

Mohammadi, Rasul (Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University)
Moussavi, Ahmad (Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University)
Zahiri, Masoome (Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.2, 2016 , pp. 381-401 More about this Journal
Abstract
According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).
Keywords
skew monoid ring; McCoy ring; strongly right AB ring; nil-reversible ring; CN ring; rings with Property (A); zip ring;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 C. Y. Hong, N. K. Kim, and Y. Lee, Extensions of McCoy's Theorem, Glasg. Math. J. 52 (2010), no. 1, 155-159.   DOI
2 C. Y. Hong, N. K. Kim, Y. Lee, and S. J. Ryu, Rings with Property (A) and their extensions, J. Algebra 315 (2007), no. 2, 612-628.   DOI
3 J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker Inc., New York, 1988.
4 J. A. Huckaba and J. M. Keller, Annihilation of ideals in commutative rings, Pacific J. Math. 83 (1979), no. 2, 375-379.   DOI
5 S. U. Hwang, N. K. Kim, and Y. Lee, On rings whose right annihilator are bounded, Glasg. Math. J. 51 (2009), no. 3, 539-559.   DOI
6 N. Jacobson, The Theory of Rings, Amer. Math. Soc., Providence, RI, 1943.
7 I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
8 D. Khurana, G. Marks, and K. Srivastava, On unit-central rings, Advances in ring theory, 205-212, Trends Math., Birkhauser/Springer Basel AG, Basel, 2010.
9 J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.
10 T. K. Lee and Y. Zhou, A unified approach to the Armendariz property of polynomial rings and power series rings, Colloq. Math. 113 (2008), no. 1, 151-169.   DOI
11 T. G. Lucas, Two annihilator conditions: Property (A) and (a.c.), Comm. Algebra 14 (1986), no. 3, 557-580.   DOI
12 G. Marks, Reversible and symmetric rings J. Pure Appl. Algebra 174 (2002), no. 3, 311-318.   DOI
13 G. Marks, R. Mazurek, and M. Zimbowski, A unified approach to various generalization of Armendariz rings Bull. Aust. Math. Soc. 81 (2010), no. 3, 361-397.   DOI
14 R. Mohammadi, A. Moussavi, and M. Zahiti, On nil-semicommutative rings, Int. Electron. J. Algebra 11 (2012), 20-37.
15 A. Moussavi and E. Hashemi, On (${\alpha}$, ${\delta}$)-skew Armendariz rings, J. Korean Math. Soc. 42 (2005), no. 2, 353-363.   DOI
16 A. R. Nasr-Isfahani and A. Moussavi, On weakly rigid rings, Glasg. Math. J. 51 (2009), no. 3, 425-440.   DOI
17 P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006), no. 1, 134-141.   DOI
18 J. Okninski, Semigroup Algebras, Marcel Dekker, New York, 1991.
19 L. Ouyang, On weak annihilator ideals of skew monoid rings, Comm. Algebra 39 (2011), no. 11, 4259-4272.   DOI
20 Y. Quentel, Sur la compacite du spectre minimal d'un anneau, Bull. Soc. Math. France 99 (1971), 265-272.
21 M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17.   DOI
22 A. B. Singh, M. R. Khan, and V. N. Dixit, Skew monoid rings over zip rings, Int. J. Algebra 4 (2010), no. 21-24, 1031-1036.
23 W. Xue, On strongly right bounded finite rings, Bull. Austral. Math. Soc. 44 (1991), no. 3, 353-355.   DOI
24 W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings, Comm. Algebra 20 (1992), no. 9, 2777-2788.   DOI
25 H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368.   DOI
26 J. M. Zelmanowitz, The finite intersection property on annihilator right ideals, Proc. Amer. Math. Soc. 57 (1976), no. 2, 213-216.   DOI
27 D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.   DOI
28 E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974), 470-473.   DOI
29 F. Azarpanah, O. A. S. Karamzadeh, and A. Rezai Aliabad, On ideals consisting entirely of zero divisors, Comm. Algebra 28 (2000), no. 2, 1061-1073.   DOI
30 J. A. Beachy and W. D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), no. 1, 1-13.   DOI
31 G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra 265 (2003), no. 2, 457-477.   DOI
32 G. F. Birkenmeier and R. P. Tucci, Homomorphic images and the singular ideal of a strongly right bounded ring, Comm. Algebra 16 (1988), no. 6, 1099-1122.   DOI
33 V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599-615.   DOI
34 J. Clark, Y. Hirano, H. K. Kim, and Y. Lee, On a generalized finite intersection property, Comm. Algebra 40 (2012), no. 6, 2151-2160.   DOI
35 P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648.   DOI
36 L. M. de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, In Proceedings of the 106th National Congress of Learned Societies, 71-73, Bibliotheque Nationale, Paris, 1982.
37 M. P. Drazin, Rings with central idempotent or nilpotent elements, Proc. Edinburgh Math. Soc. 9 (1958), no. 2, 157-165.   DOI
38 C. Faith, Algebra II, Springer-Verlag, Berlin., 1976.
39 C. Faith, Rings with zero intersection property on annihilator: zip rings, Publ. Math. 33 (1989), no. 2, 329-338.   DOI
40 C. Faith, Commutative FPF rings arising as split-null extensions, Proc. Amer. Math. Soc. 90 (1984), no. 2, 181-185.   DOI
41 C. Faith, Annihilator ideals, associated primes and Kasch-McCoy commutative rings, Comm. Algebra 19 (1991), no. 7, 1867-1892.   DOI
42 S. P. Farbman, The unique product property of groups and their amalgamated free products, J. Algebra 178 (1995), no. 3, 962-990.   DOI
43 E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. Publ. Math. 89 (1958), 79-91.   DOI
44 M. Habibi and R. Manaviyat, A generalization of nil-Armendariz rings, J. Algebra Appl. 12 (2013), no. 6, 1350001, 30 pages.
45 M. Habibi, A. Moussavi, and A. Alhevaz, The McCoy condition on ore extensions, Comm. Algebra 41 (2013), no. 1, 124-141.   DOI
46 E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 3 (2005), no. 3, 207-224.
47 M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130.   DOI
48 G. Hinkle and J. A. Huckaba, The generalized Kronecker function ring and the ring R(X), J. Reine Angew. Math. 292 (1977), 25-36.
49 Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52.   DOI
50 C. Y. Hong, N. K. Kim, T. K. Kwak, and Y. Lee, Extensions of zip rings, J. Pure Appl. Algebra 195 (2005), no. 3, 231-242.   DOI