J. L. S. Chen, N. Q. Ding's Advances in Ring Theory - Proceedings of: Proceedings of the PDF

By J. L. S. Chen, N. Q. Ding

The chosen papers during this quantity hide the entire most vital parts of ring idea and module concept corresponding to classical ring conception, illustration idea, the speculation of quantum teams, the speculation of Hopf algebras, the idea of Lie algebras and Abelian crew thought. The overview articles, written by way of experts, supply an exceptional evaluate of a number of the components of ring and module concept - excellent for researchers trying to find a brand new or comparable box of research. additionally integrated are unique articles displaying the fad of present examine.

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Hirano, M. Hongan and M. P. rings, Math. J. Okayama Univ. 24 (1982), 99-109. 24. Y. Hirano, Open Problems, International Symposium on Ring Theory (G. F. Birkenmeier, J. K. Park and Y. S. , Birkhauser, Boston (2001), 441-446. 25. S. K. Jain, P. Kanwar, S. Malik and J. B. Srivastava, KDtx is a CS-algebra, Proc. Amer. Math. Soc. 128 (2000), 397-400. 26. I. Kaplansky, Rings of Operators, Benjamin, New York, 1968. 27. T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Berlin- 25 Heidelberg-New York, 1999.

Harada) Let M = ©j € /Mj be an LE-decomposition, S = Endji(M) and J(S) denote the Jacobson radical of S. Then the following statements are equivalent. e. S is a semiregular ring. (b) Every local summand of M is a summand. (c) The decomposition complements direct summands. (d) {Mi : i 6 /} is locally semi-T-nilpotent. The proof given by Harada in [26] relies heavily on his theory of factor categories and is not for the faint-hearted. However, one can circumvent his arguments as we now explain. We first give the definition of the total of a ring, a concept pioneered by F.

Final remarks We close with two further remarks on local semi-T-nilpotency. The first is in connection with the Krull-Schmidt decomposition results which have been extensively investigated by, in particular, A. Facchini. While it has been shown that Krull-Schmidt fails for finitely presented modules over serial rings (and such modules are finite direct sums of uniserial modules) (see Chap. 9 of [19]), Dung and Facchini have shown the following in [17]. Here, for two modules A and B, we write [A]m = [B]m if there are monomorphisms from A into B and B into A while we write [A]e = [B]e if there are epimorphisms from A onto B and B onto A.

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