t-SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT DSIS A FACTORIAL DOMAIN

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

Download Full-text

Overrings of Half-Factorial Domains

Canadian Mathematical Bulletin ◽

10.4153/cmb-1994-063-2 ◽

1994 ◽

Vol 37 (4) ◽

pp. 437-442 ◽

Cited By ~ 5

Author(s):

David F. Anderson ◽

Scott T. Chapman ◽

William W. Smith

Keyword(s):

Integral Domain ◽

Dedekind Domain ◽

Irreducible Elements ◽

Factorial Domain

AbstractAn atomic integral domain D is a half-factorial domain (HFD) if for any irreducible elements α1,..., αn, β1,..., βm of D with α1... αn = β1 ...βm, then n = m. In [3], Anderson, Anderson, and Zafrullah explore factorization problems in overrings of HFDs and ask whether a localization of a HFD is again a HFD. We construct an example of a Dedekind domain which is a HFD, but with a localization which is not a HFD. We also give an example of a Dedekind domain where each localization is a HFD, but with an overring which is not a HFD.

Download Full-text

Semigroup rings as weakly factorial domains, II

International Journal of Algebra and Computation ◽

10.1142/s0218196719500085 ◽

2019 ◽

Vol 29 (03) ◽

pp. 407-418 ◽

Cited By ~ 2

Author(s):

Gyu Whan Chang ◽

Dong Yeol Oh

Keyword(s):

Quotient Group ◽

Integral Domain ◽

Semigroup Ring ◽

Semigroup Rings ◽

Type Formula ◽

Cancellative Monoid ◽

Factorial Domain

Let [Formula: see text] be an integral domain, [Formula: see text] be a nonzero torsionless commutative cancellative monoid with quotient group [Formula: see text], and [Formula: see text] be the semigroup ring of [Formula: see text] over [Formula: see text]. In this paper, among other things, we show that if [Formula: see text] (respectively, [Formula: see text], then [Formula: see text] is a weakly factorial domain if and only if [Formula: see text] is a weakly factorial GCD-domain, [Formula: see text] is a weakly factorial GCD-semigroup, and [Formula: see text] is of type [Formula: see text] (respectively, [Formula: see text] except [Formula: see text]).

Download Full-text

Elasticity in Polynomial-Type Extensions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091515000437 ◽

2016 ◽

Vol 59 (3) ◽

pp. 581-590 ◽

Cited By ~ 2

Author(s):

Mark Batell ◽

Jim Coykendall

Keyword(s):

Polynomial Ring ◽

Integral Domain ◽

Irreducible Polynomial ◽

Polynomial Rings ◽

Sufficient Condition ◽

Integral Domains ◽

Polynomial Type ◽

Factorial Domain ◽

The Relationship

AbstractThe 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 domain R and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every non-constant irreducible polynomial f ∈ R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.

Download Full-text

Linked Tori, II

10.23943/princeton/9780691166902.003.0006 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Integral Domain ◽

Quadratic Space ◽

Small Observation ◽

Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

Download Full-text

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

Jurnal Natural ◽

10.30862/jn.v8i2.340 ◽

2012 ◽

Vol 8 (2) ◽

Author(s):

Tri Widjajanti ◽

Dahlia Ramlan ◽

Rium Hilum

Keyword(s):

Polynomial Ring ◽

Integral Domain ◽

Rational Numbers ◽

Ring Of Integers ◽

Binary Operations

Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make  a construction such that any integral domain can be  a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that  f : D ® F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.

Download Full-text

Divisibility Properties of Graded Domains

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-013-3 ◽

1982 ◽

Vol 34 (1) ◽

pp. 196-215 ◽

Cited By ~ 40

Author(s):

D. D. Anderson ◽

David F. Anderson

Keyword(s):

Integral Domain ◽

Homogeneous Element ◽

Carry Over ◽

Divisibility Properties ◽

Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

Download Full-text

Decomposition and classification of length functions

Forum Mathematicum ◽

10.1515/forum-2018-0168 ◽

2020 ◽

Vol 32 (5) ◽

pp. 1109-1129

Author(s):

Dario Spirito

Keyword(s):

Integral Domain ◽

Integral Domains ◽

Bijective Correspondence ◽

Length Functions ◽

Standard Representation ◽

Prufer Domains ◽

Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

Download Full-text