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

The chosen papers during this quantity hide the entire most vital components of ring conception and module idea comparable to classical ring thought, illustration conception, the idea of quantum teams, the idea of Hopf algebras, the idea of Lie algebras and Abelian team concept. The assessment articles, written by means of experts, offer a good evaluate of many of the parts of ring and module idea - perfect for researchers searching for a brand new or similar box of research. additionally incorporated are unique articles exhibiting the rage of present study.

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We begin by recording the following definition taken from Mohamed and Miiller [42]. 1. % are summands of M with M\ + M? — M, then MI n M2 is also a summand of M. If M satisfies (Di) it is called a lifting module. A lifting module which also satisfies (£^2) is called discrete while a lifting module satisfying (Dz) is called quasi-discrete. e. every submodule of M is small. The first important result of the section establishes a nice decomposition for quasi-discrete modules due to Oshiro [45], (who called them quasisemiperfect modules).

This (somewhat verbose) phrasing emphasises that local direct summands are not always summands. It was observed by Oshiro in [46] that if every local summand of M is also a summand of M then M has an indecomposable decomposition. 17 of [42]). We now come to the interconnection between these ideas and local semiT-nilpotency. 1. However, earlier partial results appeared in papers by Yamagata [48], [49], Ishii [33], and Kanbara [36], as well as previous papers by Harada himself. 2. (Harada) Let M = ©j € /Mj be an LE-decomposition, S = Endji(M) and J(S) denote the Jacobson radical of S.

Algebra 119, 139-153 (1997). 15. Dung, N. , Modules with indecomposable decompositions that complement maximal direct summands, J. Algebra 197, 449-467, (1997). 16. Dung, N. , On the finite type of families of indecomposable modules, J. Algebra Appl. 3, 111-119 (2004). 17. Dung, N. V. , Weak Krull-Schmidt for infinite direct sums of uniserial modules, J. Algebra 193, 102-121, (1997).

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