A Note on the (Faith-Menal) Counter Example
Author(s)
R. H. Sallam
ABSTRACT
Faith-Menal counter example is an example (unique) of a right John’s ring which is not right artinian In this paper we show that the ring T which considered as an example of a right Johns ring in the (Faith-Menal) Counter Example is also artinian. The conclusion is that the unique counter example that says a right John’s ring can not be right artinian is false and the right noetherian ring with the annihilator property rl(A) = A may be artinian.
Faith-Menal counter example is an example (unique) of a right John’s ring which is not right artinian In this paper we show that the ring T which considered as an example of a right Johns ring in the (Faith-Menal) Counter Example is also artinian. The conclusion is that the unique counter example that says a right John’s ring can not be right artinian is false and the right noetherian ring with the annihilator property rl(A) = A may be artinian.
Cite this paper
R. Sallam, "A Note on the (Faith-Menal) Counter Example," Advances in Pure Mathematics, Vol. 2 No. 1, 2012, pp. 39-40. doi: 10.4236/apm.2012.21009.
R. Sallam, "A Note on the (Faith-Menal) Counter Example," Advances in Pure Mathematics, Vol. 2 No. 1, 2012, pp. 39-40. doi: 10.4236/apm.2012.21009.
References
[1] B. Johns, “Annihilator Conditions in Noetherian Rings,” Journal of Algebra, Vol. 49, No. 1, 1977, pp. 222-224. doi:10.1016/0021-8693(77)90282-4 [2] R. P. Kurshan, “Rings Whose Cyclic Modules Have Finitely Generated Socle,” Journal of Algebra, Vol. 15, No. 3, 1970, pp. 376-386. doi:10.1016/0021-8693(70)90066-9 [3] S. M. Ginn, “A Counter Example to a Theorem of KurshAn,” Journal of Algebra, Vol. 40, No. 1, 1976, pp. 105-106. doi:10.1016/0021-8693(76)90090-9 [4] W. K. Nicholson and M. F. Yousif, “Quasi-Frobenius Rings,” Series Cambridge Tracts in Mathematics, No. 158, 2003. [5] F. W. Anderson and K. R. Fuller, “Rings and Categories of Module,” Springer Verlag, New York, 1991. [6] F. Kasch, “Modules and Rings, London Mathematical Society Monographs,” Vol. 17, Academic Press, New York, 1982.
[1] B. Johns, “Annihilator Conditions in Noetherian Rings,” Journal of Algebra, Vol. 49, No. 1, 1977, pp. 222-224. doi:10.1016/0021-8693(77)90282-4 [2] R. P. Kurshan, “Rings Whose Cyclic Modules Have Finitely Generated Socle,” Journal of Algebra, Vol. 15, No. 3, 1970, pp. 376-386. doi:10.1016/0021-8693(70)90066-9 [3] S. M. Ginn, “A Counter Example to a Theorem of KurshAn,” Journal of Algebra, Vol. 40, No. 1, 1976, pp. 105-106. doi:10.1016/0021-8693(76)90090-9 [4] W. K. Nicholson and M. F. Yousif, “Quasi-Frobenius Rings,” Series Cambridge Tracts in Mathematics, No. 158, 2003. [5] F. W. Anderson and K. R. Fuller, “Rings and Categories of Module,” Springer Verlag, New York, 1991. [6] F. Kasch, “Modules and Rings, London Mathematical Society Monographs,” Vol. 17, Academic Press, New York, 1982.