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

The chosen papers during this quantity hide all of the most crucial parts of ring concept and module thought resembling classical ring idea, illustration thought, the idea of quantum teams, the idea of Hopf algebras, the idea of Lie algebras and Abelian staff thought. The evaluation articles, written by means of experts, supply an exceptional assessment of many of the parts of ring and module idea - excellent for researchers trying to find a brand new or similar box of analysis. additionally integrated are unique articles exhibiting the rage of present study.

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**Extra resources for Advances in Ring Theory - Proceedings of: Proceedings of the 4th China-Japan-Korea International Conference, Nanjing, China 24-28 June 2004**

**Example text**

B) ® jgtfMj is an extending module for every countable subset H of I. (c) Mi (B MJ is an extending module for every distinct pair of indices i,j € / and {Mi : i € /} is IsTn and satisfies (A%). In this case, any local summand of M is a summand. 2 also comes from Harada and Y. Oshiro's [29] and M. A. Kamal and Miiller's [35]. We also note that, with the same hypothesis as Theorem 12, further characterizations of when M is extending are given by J. Kado, Y. ) and the concepts of generalized injectivity and the internal exchange property.

Obviously, E(G) is a subnear-ring of (M0(G), +, •). Thus we say that E(G) is the endomorphism near-ring of the group G. For the remainder basic concepts and results on ring and near-ring case, we refer to [1], [8] and [9]. 2. R-groups. R-group, and then investigate their properties. For any ring R, right E-modules M and N, the set of all /^-module homomorphisms from M to N is denoted by HorriR(M,N) and the set of all group homomorphisms from M to N is Hom(M,N):=Homz(M,N), in particular we denote that Endn(M} := HortiR(M,M) and End(M) := Endz(M), In this case, M is called a GM module over R if every group homomorphism of M is an R-module homomorphism, that is, End(M) = EndR(M).

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.