By Sergio R. López-Permouth, Dinh Van Huynh
This quantity comprises refereed examine and expository articles via either plenary and different audio system on the foreign convention on Algebra and purposes held at Ohio college in June 2008, to honor S.K. Jain on his seventieth birthday. The articles are on a large choice of components in classical ring conception and module concept, equivalent to jewelry gratifying polynomial identities, earrings of quotients, workforce earrings, homological algebra, injectivity and its generalizations, and so forth. integrated also are functions of ring thought to difficulties in coding thought and in linear algebra.
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Additional info for Advances in Ring Theory (Trends in Mathematics)
Sample text
We have that sext (R) is the class of all modules closed under extensions generated by the left ideals of R. Since R is left noetherian, then each left ideal is finitely generated, thus every left ideal of R belongs to qext (R). As qext (R) is the class of all finitely generated left R-modules, then sext (R) ⊆ qext (R). The other inclusion follows from the fact that, for each finitely generated module R M , qext (M ) = qext (S), where S is the unique simple module in R-simp. Indeed M = socn (M ) ∈ sext (S) for some n ∈ N.
Thus the right perfect factors of R are left semiartinian and left noetherian, thus they are left artinian. 3] and by straightforward uses of Morita equivalence theory. 6. If R is a left artinian, local ring such that E (R/ Rad (R)) is finitely generated, then sext (R) = qext (R) . Proof. We have that sext (R) is the class of all modules closed under extensions generated by the left ideals of R. Since R is left noetherian, then each left ideal is finitely generated, thus every left ideal of R belongs to qext (R).
References [1] A. Alvarado, H. Rinc´ on and J. R´ıos, On the lattices of natural and conatural classes in R-mod, Comm. Algebra. 29 (2) (2001) 541–556. [2] A. Alvarado, H. Rinc´ on and J. R´ıos, On Some Lattices of Module Classes, Journal of Algebra and its Applications. 2006, 105–117. [3] L. Bican, T. Kepka, P. N˘emec, Rings, modules, and preradicals. Lecture Notes in Pure and Applied Mathematics, 75. , New York, 1982. [4] J. Dauns, Y. Zhou, Classes of Modules. Pure and Applied Mathematics (Boca Raton), 281.