If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. ... more If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.
An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for... more An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x ∈ M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, lengthfactoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is in general weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M , and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between lengthfactoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely short and a purely long irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.
In this paper we generalize the standard notion of "unique factorization domains" (UFDs) to the t... more In this paper we generalize the standard notion of "unique factorization domains" (UFDs) to the the nonatomic situation. The main result of this paper is that, in contrast to the atomic situation, the assumption that every irreducible is prime (AP) and the notion that every (atomic) nonzero nonunit can be factored uniquely into irreducible elements are distinct notions.
This paper investigates a strong convergence property of rings of formal power series where the c... more This paper investigates a strong convergence property of rings of formal power series where the coefficient ring has the strong finite type (SFT) property. In particular, we show that if R[[x]] is SFT, then given two primes Γ0⊂6=Γ1 whose variable annihilation ideals agree, then x ∈ Γ1. We also show that this strong convergence property completely characterizes Noetherian rings. 1.
Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whet... more Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x; M ] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M = N 0 : he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F [x; M ] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x; M ] is not atomic for any integral domain R. Then for every n ≥ 2 and for each field F of finite characteristic we exhibit a torsion-free atomic monoid of rank n such that F [x; M ] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z 2 [x; M ] is not atomic.
In this paper, we show that it is possible for a commutative ring with identity to be non-atomic ... more In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In particular, we produce a commutative ring with identity, R, that is antimatter (that is, R has no irreducibles whatsoever) such that R[t] is strongly atomic. What is more, given any nonzero nonunit f (t) ∈ R[t] then there is a factorization of f (t) into irreducibles of length no more than deg(f (t)) + 2.
Page 111. Chapter 5 HALF-FACTORIAL DOMAINS, A SURVEY Scott T. Chapman Department of Mathematics T... more Page 111. Chapter 5 HALF-FACTORIAL DOMAINS, A SURVEY Scott T. Chapman Department of Mathematics Trinity University 715 Stadium Drive San Antonio, Texas 78212-7200 schapman@ trinity. edu Jim Coykendall Department ...
The SFT (for strong finite type) condition was introduced by J. Arnold [1] in the context of stud... more The SFT (for strong finite type) condition was introduced by J. Arnold [1] in the context of studying the condition for the formal power series rings to have finite dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (resp. ring) that has the VSFT property has the SFT property. In this paper we explore the interplay of the SFT property and the VSFT property.
Proceedings of the Edinburgh Mathematical Society, 2016
The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is... more The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domainRand the elasticity of its polynomial ringR[x]. For example, ifRhas at least one atom, a sufficient condition for the polynomial ringR[x] to have elasticity 1 is that every non-constant irreducible polynomialf∈R[x] be irreducible inK[x]. We will determine the integral domainsRwhose polynomial rings satisfy this condition.
In this note, we use the A + XB[X] and A + XI[X] constructions from a new angle to construct new ... more In this note, we use the A + XB[X] and A + XI[X] constructions from a new angle to construct new examples of half factorial domains. Positive results are obtained highlighting the interplay between the notions of integrally closed domain and half-factorial domain in A + XB[X] constructions. It is additionally shown that constructions of the form A + XI[X] rarely possess the half-factorial property.
In this note, we use the A+XB(X) and A+XI(X) constructions from a new angle to construct new exam... more In this note, we use the A+XB(X) and A+XI(X) constructions from a new angle to construct new examples of half factorial domains. Pos- itive results are obtained highlighting the interplay between the notions of integrally closed domain and half-factorial domain in A + XB(X) construc- tions. It is additionally shown that constructions of the form A + XI(X) rarely possess the half-factorial property.
ABSTRACT We survey the research conducted on zero divisor graphs, with a focus on zero divisor gr... more ABSTRACT We survey the research conducted on zero divisor graphs, with a focus on zero divisor graphs determined by equivalence classes of zero divisors of a commutative ring R. In particular, we consider the problem of classifying star graphs with any finite number of vertices. We study the pathology of a zero divisor graph in terms of cliques, we investigate when the clique and chromatic numbers are equal, and we show that the girth of a Noetherian ring, if finite, is 3. We also introduce a graph for modules that is useful for studying zero divisor graphs of trivial extensions.
It is well known that the factorization properties of a domain are reflected in the structure of ... more It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.
This collection of papers in commutative algebra stemmed out of the 2009 Fall Southeastern Americ... more This collection of papers in commutative algebra stemmed out of the 2009 Fall Southeastern American Mathematical Society Meeting which contained three special sessions in the field: Special Session on Commutative Ring Theory, a Tribute to the Memory of James Brewer, organized by Alan Loper and Lee Klingler; Special Session on Homological Aspects of Module Theory, organized by Andy Kustin, Sean Sather-Wagstaff, and Janet Vassilev; and Special Session on Graded Resolutions, organized by Chris Francisco and Irena Peeva. Much of the commutative algebra community has split into two camps, for lack of a better word: the Noetherian camp and the non-Noetherian camp. Most researchers in commutative algebra identify with one camp or the other, though there are some notable exceptions to this. We had originally intended this to be a Proceedings Volume for the conference as the sessions had a nice combination of both Noetherian and non-Noetherian talks. However, the project grew into two Volumes with invited papers that are blends of survey material and new research. We hope that members from the two camps will read each others' papers and that this will lead to increased mathematical interaction between the camps. As the title suggests, this volume, Progress in Commutative Algebra II, contains surveys on aspects of closure operations, finiteness conditions and factorization. Contributions to this volume have come mainly from speakers in the first and second sessions and from invited articles on closure operations, test ideals, Noetherian rings without finite normaliztion and non-unique factorization. The collection documents some current trends in two of the most active areas of commutative algebra. Closure operations on ideals and modules are a bridge between Noetherian and non-Noetherian commutative algebra. The Noetherian camp typically study structures related to a particular closure operation such as the core or the test ideal or how particular closure operations yield nice proofs of hard theorems. The non-Noetherian camp approach closure operations from the view of multiplicative ideal theory and the relationship to Kronecker function rings. This volume contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure. Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. For instance, the division between the Noetherian and the vi Preface non-Noetherian crowd comes down to the property that all ideals in a Noetherian ring are finitely generated, by definition. However, in the study of non-Noetherian rings it is much easier to find a ring having a finite number of prime ideals. We have included papers by Boynton and Sather-Wagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely generated. The final three papers in this volume investigate factorization in a broad sense. The first paper by Celikbas and Eubanks-Turner discusses the partially ordered set of prime ideals of the projective line over the integers. We have also included a paper on zero divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and Spiroff. The final paper, by Chapman and Krause, concerns non-unique factorization. The first session was a Tribute to the Memory of James Brewer. As many of the authors participated in this session, we dedicate this volume to Brewer's memory. Enjoy!
We consider certain pullback constructions in the spirit of Int(E; D). It is well-known that if E... more We consider certain pullback constructions in the spirit of Int(E; D). It is well-known that if E is a nite subset of D, then Int(E; D) is non-atomic. Since Int(E; D) may be de ned by means of a conductor square, it is natural to ask if this non-atomicity property exists in a more general setting. The authors show that although non-atomicity is usually to be expected in the more general case, certain restrictive conditions do necessitate atomicity. As a consequence of the main result, we have an atomic domain that does not satisfy the ascending chain condition on principal ideals.
If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. ... more If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.
An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for... more An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x ∈ M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, lengthfactoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is in general weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M , and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between lengthfactoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely short and a purely long irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.
In this paper we generalize the standard notion of "unique factorization domains" (UFDs) to the t... more In this paper we generalize the standard notion of "unique factorization domains" (UFDs) to the the nonatomic situation. The main result of this paper is that, in contrast to the atomic situation, the assumption that every irreducible is prime (AP) and the notion that every (atomic) nonzero nonunit can be factored uniquely into irreducible elements are distinct notions.
This paper investigates a strong convergence property of rings of formal power series where the c... more This paper investigates a strong convergence property of rings of formal power series where the coefficient ring has the strong finite type (SFT) property. In particular, we show that if R[[x]] is SFT, then given two primes Γ0⊂6=Γ1 whose variable annihilation ideals agree, then x ∈ Γ1. We also show that this strong convergence property completely characterizes Noetherian rings. 1.
Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whet... more Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x; M ] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M = N 0 : he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F [x; M ] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x; M ] is not atomic for any integral domain R. Then for every n ≥ 2 and for each field F of finite characteristic we exhibit a torsion-free atomic monoid of rank n such that F [x; M ] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z 2 [x; M ] is not atomic.
In this paper, we show that it is possible for a commutative ring with identity to be non-atomic ... more In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In particular, we produce a commutative ring with identity, R, that is antimatter (that is, R has no irreducibles whatsoever) such that R[t] is strongly atomic. What is more, given any nonzero nonunit f (t) ∈ R[t] then there is a factorization of f (t) into irreducibles of length no more than deg(f (t)) + 2.
Page 111. Chapter 5 HALF-FACTORIAL DOMAINS, A SURVEY Scott T. Chapman Department of Mathematics T... more Page 111. Chapter 5 HALF-FACTORIAL DOMAINS, A SURVEY Scott T. Chapman Department of Mathematics Trinity University 715 Stadium Drive San Antonio, Texas 78212-7200 schapman@ trinity. edu Jim Coykendall Department ...
The SFT (for strong finite type) condition was introduced by J. Arnold [1] in the context of stud... more The SFT (for strong finite type) condition was introduced by J. Arnold [1] in the context of studying the condition for the formal power series rings to have finite dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (resp. ring) that has the VSFT property has the SFT property. In this paper we explore the interplay of the SFT property and the VSFT property.
Proceedings of the Edinburgh Mathematical Society, 2016
The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is... more The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domainRand the elasticity of its polynomial ringR[x]. For example, ifRhas at least one atom, a sufficient condition for the polynomial ringR[x] to have elasticity 1 is that every non-constant irreducible polynomialf∈R[x] be irreducible inK[x]. We will determine the integral domainsRwhose polynomial rings satisfy this condition.
In this note, we use the A + XB[X] and A + XI[X] constructions from a new angle to construct new ... more In this note, we use the A + XB[X] and A + XI[X] constructions from a new angle to construct new examples of half factorial domains. Positive results are obtained highlighting the interplay between the notions of integrally closed domain and half-factorial domain in A + XB[X] constructions. It is additionally shown that constructions of the form A + XI[X] rarely possess the half-factorial property.
In this note, we use the A+XB(X) and A+XI(X) constructions from a new angle to construct new exam... more In this note, we use the A+XB(X) and A+XI(X) constructions from a new angle to construct new examples of half factorial domains. Pos- itive results are obtained highlighting the interplay between the notions of integrally closed domain and half-factorial domain in A + XB(X) construc- tions. It is additionally shown that constructions of the form A + XI(X) rarely possess the half-factorial property.
ABSTRACT We survey the research conducted on zero divisor graphs, with a focus on zero divisor gr... more ABSTRACT We survey the research conducted on zero divisor graphs, with a focus on zero divisor graphs determined by equivalence classes of zero divisors of a commutative ring R. In particular, we consider the problem of classifying star graphs with any finite number of vertices. We study the pathology of a zero divisor graph in terms of cliques, we investigate when the clique and chromatic numbers are equal, and we show that the girth of a Noetherian ring, if finite, is 3. We also introduce a graph for modules that is useful for studying zero divisor graphs of trivial extensions.
It is well known that the factorization properties of a domain are reflected in the structure of ... more It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.
This collection of papers in commutative algebra stemmed out of the 2009 Fall Southeastern Americ... more This collection of papers in commutative algebra stemmed out of the 2009 Fall Southeastern American Mathematical Society Meeting which contained three special sessions in the field: Special Session on Commutative Ring Theory, a Tribute to the Memory of James Brewer, organized by Alan Loper and Lee Klingler; Special Session on Homological Aspects of Module Theory, organized by Andy Kustin, Sean Sather-Wagstaff, and Janet Vassilev; and Special Session on Graded Resolutions, organized by Chris Francisco and Irena Peeva. Much of the commutative algebra community has split into two camps, for lack of a better word: the Noetherian camp and the non-Noetherian camp. Most researchers in commutative algebra identify with one camp or the other, though there are some notable exceptions to this. We had originally intended this to be a Proceedings Volume for the conference as the sessions had a nice combination of both Noetherian and non-Noetherian talks. However, the project grew into two Volumes with invited papers that are blends of survey material and new research. We hope that members from the two camps will read each others' papers and that this will lead to increased mathematical interaction between the camps. As the title suggests, this volume, Progress in Commutative Algebra II, contains surveys on aspects of closure operations, finiteness conditions and factorization. Contributions to this volume have come mainly from speakers in the first and second sessions and from invited articles on closure operations, test ideals, Noetherian rings without finite normaliztion and non-unique factorization. The collection documents some current trends in two of the most active areas of commutative algebra. Closure operations on ideals and modules are a bridge between Noetherian and non-Noetherian commutative algebra. The Noetherian camp typically study structures related to a particular closure operation such as the core or the test ideal or how particular closure operations yield nice proofs of hard theorems. The non-Noetherian camp approach closure operations from the view of multiplicative ideal theory and the relationship to Kronecker function rings. This volume contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure. Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. For instance, the division between the Noetherian and the vi Preface non-Noetherian crowd comes down to the property that all ideals in a Noetherian ring are finitely generated, by definition. However, in the study of non-Noetherian rings it is much easier to find a ring having a finite number of prime ideals. We have included papers by Boynton and Sather-Wagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely generated. The final three papers in this volume investigate factorization in a broad sense. The first paper by Celikbas and Eubanks-Turner discusses the partially ordered set of prime ideals of the projective line over the integers. We have also included a paper on zero divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and Spiroff. The final paper, by Chapman and Krause, concerns non-unique factorization. The first session was a Tribute to the Memory of James Brewer. As many of the authors participated in this session, we dedicate this volume to Brewer's memory. Enjoy!
We consider certain pullback constructions in the spirit of Int(E; D). It is well-known that if E... more We consider certain pullback constructions in the spirit of Int(E; D). It is well-known that if E is a nite subset of D, then Int(E; D) is non-atomic. Since Int(E; D) may be de ned by means of a conductor square, it is natural to ask if this non-atomicity property exists in a more general setting. The authors show that although non-atomicity is usually to be expected in the more general case, certain restrictive conditions do necessitate atomicity. As a consequence of the main result, we have an atomic domain that does not satisfy the ascending chain condition on principal ideals.
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Papers by Jim Coykendall