Good Conditions for the Total
Let R be a ring with 1 ∈ R. In Mod-R the total Tot(M, N) is a semi-ideal, which contains the radical Rad(M, N), the singular ideal △(M,N)and the cosingular ideal ▽(M, N). We study conditions on modules Q and P, which imply that Rad(Q, N) = △(Q,N) = Tot(Q, N) and Rad(M, P) = ▽(M, P) = Tot(M, P) for all M and N. We prove that these equalities hold if Q is injective, resp. P is semi-perfect and projective. To get further results and interesting topics, we consider the question: For which Q is △(Q,N) = Tot(Q, N) for all N? Here we study rings R such that the condition △(Q, N) = Tot(Q, N) for all N implies that Q is a direct sum of injective modules. We conjecture that such rings must be right Noetherian and prove that they (and all their homomorphic images) are right Goldie rings. Further, the conjecture is confirmed in a number of cases.
Unable to display preview. Download preview PDF.
- M. F. Atyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, New York, 1969.Google Scholar
- K. I. Beidar and R. Wiegandt, Radicals induced by the total of rings, ibid., 149–159.Google Scholar
- A. W. Chatter and C. R. Hajarnavis, Rings with Chain Conditions, Pitman, Boston, 1980.Google Scholar
- C. Faith, Algebra I. Rings, Modules, and Categories, SpringerVerlag, Heidelberg, New York, 1976.Google Scholar
- F. Kasch, Moduln mit LE-Zerlegung und Harada-Moduln, Lecture Notes, München, 1982.Google Scholar
- F. Kasch, The total in the category of modules, General Algebra (1988), 129–137, Elsevier Sci. Pub., Amsterdam.Google Scholar
- F. Kasch and W. Schneider, Exchange properties and the total, Advances in Ring Theory, edited by S. K. Jain and S. Tariq Rizvi, Birkhäuser, Boston, 1997, 163–174.Google Scholar
- G. O. Michler, Prime right ideals and right Noetherian rings, in Ring Theory, edited by R. Gordon, Academic Press, New York, (1972), 251–255.Google Scholar