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D. D. Anderson

Bio: D. D. Anderson is an academic researcher from University of Iowa. The author has contributed to research in topics: Commutative ring & Integral domain. The author has an hindex of 36, co-authored 182 publications receiving 4312 citations. Previous affiliations of D. D. Anderson include Indiana University of Pennsylvania & Virginia Tech.


Papers
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Journal ArticleDOI
TL;DR: In this paper, it was shown that if A is a regular Noetherian ring with maximal ideals N 1,..., Ns, such that each A/Ni is finite, then for R = A/Nn11 ··· Nnss, χ(R) = cl(R).

331 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied factorization in an integral domain R, that is, factoring elements of R into products of irreducible elements, and investigated several factorization properties in R which are weaker than unique factorization.

272 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that R is Armendariz if and only if each homomorphic image of R is a homomorphism of R. Theorem 5.1.
Abstract: We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a i b j=0 for each coefficient a i of f and b j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.

262 citations

Journal ArticleDOI
TL;DR: In this paper, a ring is a clean ring if every element of a ring can be written uniquely as the sum of a unit and an idempotent, which is the definition of uniquely clean rings.
Abstract: As defined by Nicholson a (noncommutative) ring is a clean ring if every element of is a sum of a unit and an idempotent. Let be a commutative ring with identity. We define to be a uniquely clean ring if every element of can be written uniquely as the sum of a unit and an idempotent. Examples of clean rings (uniquely clean rings) include von Neumann regular rings (Boolean rings) and quasilocal rings (with residue field ). A ring is a clean ring or uniquely clean ring if and only if is. So every zero-dimensional ring is a clean ring, but a zero-dimensional ring is a uniquely clean ring if and only if is a Boolean ring.

132 citations


Cited by
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Reference BookDOI
11 May 2018
TL;DR: In this paper, a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work, is presented and accompanied by complete proofs, where the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category.
Abstract: 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.

1,141 citations

Journal ArticleDOI
TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).

1,087 citations

Journal ArticleDOI
TL;DR: The second edition of the ONAG book as mentioned in this paper presents recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.
Abstract: ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

605 citations

Book ChapterDOI
25 Sep 2007

425 citations

Journal ArticleDOI
Jay A. Wood1
TL;DR: In this paper, the authors set a foundation for the study of linear codes over finite Frobenius rings, which is the most appropriate for coding theoretic purposes because two classical theorems of Mac Williams, the extension theorem and the Mac Williams identities, generalize from finite fields to finite rings.
Abstract: This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of Mac Williams, the extension theorem and the Mac Williams identities, generalize from finite fields to finite Frobenius rings. It is over Frobenius rings that certain key identifications can be made between the ring and its complex characters.

373 citations