This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Foundations of Module and Ring Theory : A Handbook for Study and Research

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

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Integral closure of ideals, rings, and modules

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Infinite Abelian Groups

Factorization in integral domains

Coefficients and values Additive structure Stone-Weierstrass Integer-valued polynomials on a subset Prime ideals Multiplicative properties Skolem properties Invertible ideals and the Picard group Integer-valued derivatives and finite differences Integer-valued rational functions Integer-valued polynomials in several indeterminates References List of symbols Index.

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Integer-Valued Polynomials

Ratliff and Rush show in particular that Ĩ is the largest ideal for which, for sufficiently large positive integers n, (Ĩ) = I and hence that ̃̃ I = Ĩ. We call regular ideals I for which I = Ĩ Ratliff–Rush ideals, and we call Ĩ the Ratliff–Rush ideal associated with I. It is easy to see that an element a of I : I is integral over I, in the sense that there is an equation of the form a + b1a k−1 + . . . + bk = 0, where bi ∈ I for i = 1, . . . , k. Therefore, the ideal Ĩ is always between I and the integral closure I ′ of I, and hence integrally closed ideals are Ratliff–Rush ideals. Ratliff and Rush observe [RR, (2.3.4)] that the powers of an invertible ideal are Ratliff–Rush ideals, so any principal ideal generated by a nonzerodivisor is a Ratliff–Rush ideal. They also prove the interesting fact that for any regular ideal I of R, there is a positive integer n such that for all k ≥ n, Ĩk = I [RR, (2.3.2)], i.e., all sufficiently high powers of a regular ideal are Ratliff–Rush. A regular ideal I is always a reduction of its associated Ratliff–Rush ideal Ĩ, in the sense that I(Ĩ) = (Ĩ) for some positive integer n. For the basic facts on reductions and reduction numbers of ideals, we refer the reader to [NR], [H1], and [H2]. In particular, if there is an element a of an ideal I for which aI = I then aR is called a principal reduction of I and the smallest n for which this equation holds is called the reduction number of I. We will call a regular ideal I stable iff it has a principal reduction with reduction number at most one, i.e., iff there is an element a of I for which

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The ratliff-rush ideals in a noetherian ring

Coefficient Ideals in and Blowups of a Commutative Noetherian Domain

The Laskerian property in commutative rings

Commutative rings with acc on n-generated ideals

We describe an example showing that ACCP need not extend from a ring to a polynomial ring over it.

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David Lantz

Papers

The ratliff-rush ideals in a noetherian ring

Coefficient Ideals in and Blowups of a Commutative Noetherian Domain

The Laskerian property in commutative rings

Commutative rings with acc on n-generated ideals

Accp in polynomial rings: a counterexample