A Note on the (Faith-Menal) Counter Example ()
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.
Share and Cite:
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.
Conflicts of Interest
The authors declare no conflicts of interest.
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.
|