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

ISBN-10: 981256425X

ISBN-13: 9789812564252

The chosen papers during this quantity conceal the entire most vital components of ring conception and module thought comparable to classical ring idea, illustration thought, the idea of quantum teams, the idea of Hopf algebras, the idea of Lie algebras and Abelian crew conception. The overview articles, written by way of experts, supply a good evaluation of some of the components of ring and module concept - perfect for researchers trying to find a brand new or comparable box of analysis. additionally integrated are unique articles displaying the rage of present examine.

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2 is superfluous. We mention one further result on quasi-discrete modules from Mohamed and Miiller [42]. 3. If M = ©jg/M,- is a direct sum of pairwise mutually projective local modules Mi, the following statements are equivalent. (a) (b) (c) (d) (e) M is quasi-discrete. M has small radical. Every proper submodule of M is contained in a maximal submodule. M = ©jg/Mj complements direct summands. {Mi : i € 1} is locally semi- T-nilpotent. We next note that J. M. Zelmanowitz [50] gives a quick proof that if M is a discrete module with endomorphism ring 5 then S/J(S) is isomorphic to a direct product of endomorphism rings of vector spaces over division rings (cf.

K. R. Goodearl, Simple Noetherian rings not isomorphic to matrix rings over domains, Comm. Algebra 12 (1984), 1412-1434. 23. Y. 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.

B) The decomposition complements direct summands. (c) {Mi : i £ 1} is locally semi-T-nilpotent. We note that D. Khurana and R. N. Gupta [40] give an alternative approach to identifying J(S) and S/J(S) on the assumption that the LEdecomposition is IsTn. Next we state a theorem due to N. V. Dung [15]. This result is important for two reasons. Firstly, it generalises part of Harada's Theorem by replacing the LE-decomposition hypothesis by the weaker complementing maximal summands condition. Secondly, his method of proof is quite different from that of Harada's, employing module theory techniques similar to those used in a paper by Zimmermann-Huisgen and Zimmermann [51] which we will feature later.

### Advances in Ring Theory by J. L. S. Chen, N. Q. Ding

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