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ON ANNIHILATIONS OF IDEALS IN SKEW MONOID RINGS
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 Title & Authors
ON ANNIHILATIONS OF IDEALS IN SKEW MONOID RINGS
Mohammadi, Rasul; Moussavi, Ahmad; Zahiri, Masoome;
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 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 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 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;
 Language
English
 Cited by
 References
1.
D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272. crossref(new window)

2.
E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974), 470-473. crossref(new window)

3.
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. crossref(new window)

4.
J. A. Beachy and W. D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), no. 1, 1-13. crossref(new window)

5.
H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. crossref(new window)

6.
G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra 265 (2003), no. 2, 457-477. crossref(new window)

7.
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. crossref(new window)

8.
V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599-615. crossref(new window)

9.
J. Clark, Y. Hirano, H. K. Kim, and Y. Lee, On a generalized finite intersection property, Comm. Algebra 40 (2012), no. 6, 2151-2160. crossref(new window)

10.
P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648. crossref(new window)

11.
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.

12.
M. P. Drazin, Rings with central idempotent or nilpotent elements, Proc. Edinburgh Math. Soc. 9 (1958), no. 2, 157-165. crossref(new window)

13.
C. Faith, Algebra II, Springer-Verlag, Berlin., 1976.

14.
C. Faith, Commutative FPF rings arising as split-null extensions, Proc. Amer. Math. Soc. 90 (1984), no. 2, 181-185. crossref(new window)

15.
C. Faith, Rings with zero intersection property on annihilator: zip rings, Publ. Math. 33 (1989), no. 2, 329-338. crossref(new window)

16.
C. Faith, Annihilator ideals, associated primes and Kasch-McCoy commutative rings, Comm. Algebra 19 (1991), no. 7, 1867-1892. crossref(new window)

17.
S. P. Farbman, The unique product property of groups and their amalgamated free products, J. Algebra 178 (1995), no. 3, 962-990. crossref(new window)

18.
E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. Publ. Math. 89 (1958), 79-91. crossref(new window)

19.
M. Habibi and R. Manaviyat, A generalization of nil-Armendariz rings, J. Algebra Appl. 12 (2013), no. 6, 1350001, 30 pages.

20.
M. Habibi, A. Moussavi, and A. Alhevaz, The McCoy condition on ore extensions, Comm. Algebra 41 (2013), no. 1, 124-141. crossref(new window)

21.
E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 3 (2005), no. 3, 207-224.

22.
M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130. crossref(new window)

23.
G. Hinkle and J. A. Huckaba, The generalized Kronecker function ring and the ring R(X), J. Reine Angew. Math. 292 (1977), 25-36.

24.
Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52. crossref(new window)

25.
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. crossref(new window)

26.
C. Y. Hong, N. K. Kim, and Y. Lee, Extensions of McCoy's Theorem, Glasg. Math. J. 52 (2010), no. 1, 155-159. crossref(new window)

27.
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. crossref(new window)

28.
J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker Inc., New York, 1988.

29.
J. A. Huckaba and J. M. Keller, Annihilation of ideals in commutative rings, Pacific J. Math. 83 (1979), no. 2, 375-379. crossref(new window)

30.
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. crossref(new window)

31.
N. Jacobson, The Theory of Rings, Amer. Math. Soc., Providence, RI, 1943.

32.
I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.

33.
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.

34.
J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.

35.
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. crossref(new window)

36.
T. G. Lucas, Two annihilator conditions: Property (A) and (a.c.), Comm. Algebra 14 (1986), no. 3, 557-580. crossref(new window)

37.
G. Marks, Reversible and symmetric rings J. Pure Appl. Algebra 174 (2002), no. 3, 311-318. crossref(new window)

38.
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. crossref(new window)

39.
R. Mohammadi, A. Moussavi, and M. Zahiti, On nil-semicommutative rings, Int. Electron. J. Algebra 11 (2012), 20-37.

40.
A. Moussavi and E. Hashemi, On (${\alpha}$, ${\delta}$)-skew Armendariz rings, J. Korean Math. Soc. 42 (2005), no. 2, 353-363. crossref(new window)

41.
A. R. Nasr-Isfahani and A. Moussavi, On weakly rigid rings, Glasg. Math. J. 51 (2009), no. 3, 425-440. crossref(new window)

42.
P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006), no. 1, 134-141. crossref(new window)

43.
J. Okninski, Semigroup Algebras, Marcel Dekker, New York, 1991.

44.
L. Ouyang, On weak annihilator ideals of skew monoid rings, Comm. Algebra 39 (2011), no. 11, 4259-4272. crossref(new window)

45.
Y. Quentel, Sur la compacite du spectre minimal d'un anneau, Bull. Soc. Math. France 99 (1971), 265-272.

46.
M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17. crossref(new window)

47.
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.

48.
W. Xue, On strongly right bounded finite rings, Bull. Austral. Math. Soc. 44 (1991), no. 3, 353-355. crossref(new window)

49.
W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings, Comm. Algebra 20 (1992), no. 9, 2777-2788. crossref(new window)

50.
J. M. Zelmanowitz, The finite intersection property on annihilator right ideals, Proc. Amer. Math. Soc. 57 (1976), no. 2, 213-216. crossref(new window)