The group of divisibility and its applications

1973 ◽  
pp. 194-208 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Group Of Divisibility

scholarly journals On the Graph of Divisibility of an Integral Domain

2015 ◽  
Vol 58 (3) ◽  
pp. 449-458 ◽  
Author(s):  
Jason Greene Boynton ◽  
Jim Coykendall
Keyword(s):  
Topological Space ◽  
Integral Domain ◽  
Point Of View ◽  
Current Understanding ◽  
Main Theme ◽  
Topological Graph ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Theoretic Point ◽  
Group Of Divisibility

AbstractIt 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.

Convex Directed Subgroups of a Group of Divisibility

1974 ◽  
Vol 26 (3) ◽  
pp. 532-542 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Integral Domain ◽  
Multiplicative Group ◽  
Ordered Group ◽  
Quotient Field ◽  
Partially Ordered ◽  
Group Of Units ◽  
Partially Ordered Group ◽  
Group Of Divisibility

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

Semi-Valuations and Groups of Divisibility

1969 ◽  
Vol 21 ◽  
pp. 576-591 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Theoretic Problem ◽  
Group Of Divisibility

Associated with any integral domain R there is a partially ordered group A, called the group of divisibility of R. When R is a valuation ring, A is merely the value group; and in this case, ideal-theoretic properties of R are easily derived from corresponding properties of A, and conversely. Even in the general case, though, it has proved useful on occasion to phrase a ring-theoretic problem in terms of the ordered group A, first solve the problem there, and then pull back the solution if possible to R. Lorenzen (15) originally applied this technique to solve a problem of Krull, and Nakayama (16) used it to produce a counterexample to another question of Krull. More recently, Heinzer (7;8) has used the method to construct other interesting examples of rings.

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