On algebraic closures

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 Enochs and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic). Later work and applications are given

Arithmetic-arboreal residue structures induced by Prufer extensions : An axiomatic approach

We present an axiomatic framework for the residue structures induced by Prufer extensions with a stress upon the intimate connection between their arithmetic and arboreal theoretic properties. The main result of the paper provides an adjunction relationship between two naturally defined functors relating Prufer extensions and superrigid directed commutative regular quasi-semirings.Comment: 56 page

Smarandache rings

Over the past 25 years, I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings) specially to motivate both ring theorists and Smarandache algebraists to develop and study several important and innovative properties about S-rings

The residue fields of a zero-dimensional ring

AbstractGilmer and Heinzer have considered the question: For an indexed family of fields oK = {Kα}αgEA, under what conditions does there exist a zero-dimensional ring R (always commutative with unity) such that oK is up to isomorphism the family of residue fields {RMα}αgEA of R? If oK is the family of residue fields of a zero-dimensional ring R, then the associated bijection from the index set A to the spectrum of R (with the Zariski topology) gives A the topology of a Boolean space. The present paper considers the following question: Given a field F, a Boolean space X and a family {Kx}xgEX of extension fields of F, under what conditions does there exist a zero-dimensional F-algebra R such that oK is up to F-isomorphism the family of residue fields of R and the associated bijection from X to Spec(R) is a homeomorphism? A necessary condition is that given x in X and any finite extension E of F in Kx, there exist a neighborhood V of x and, for each y in V, an F-embedding of E into Ky. We prove several partial converses of this result, under hypotheses which allow the “straightening” of the F-embeddings to make them compatible. We give particular attention to the cases where X has only one accumulation point and where X is countable; and we provide several examples