We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an degree extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.
In this article, we introduce a special class of non complete webs, the NN-webs. We also study the algebraic and geometric properties of these webs.
This is a description of some different approaches which have been taken to the problem of generalizing the algebraic closure of a field. Work surveyed is by Enoch and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic).Later work and applications are given.
Let be the ring of real continuous functions on a completely regular Hausdorff space. In this paper an almost discrete space is determined by the algebraic structure of . The intersection of essential weak ideal in is also studied.
Primary and secondary functors have been introduced in [2] and applied to extend some results concerning asymptotic prime ideals. In this paper, the theory of primary and secondary functors is developed and examples of non-exact primary and non-exact secondary functors are presented. Also, as an application, the sets of associated and of attached prime ideals of certain modules are determined.