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Paper   IPM / M / 16529
School of Mathematics
  Title:   Substructures with an almost division algorithm in Euclidean commutator lattices
  Author(s):  Amir Masoud Rahimi (Joint with E. Mehdi-Nezhad)
  Status:   To Appear
  Journal: Palestine J. Math.
  Supported by:  IPM
In this article we introduce the notion of a Euclidean commutator lattice L and study some of its algebraic properties in a parallel fashion with Euclidean (rings, semirings, modules, and semimodules). The notion of an almost division algorithm and join absorptive subsets of order t ≥ 1 (t a fixed integer) in L as a parallel extension of additively absorptive subsemirings of Euclidean semirings are discussed. For a fixed integer t ≥ 1, a lower set D of a Euclidean commutator lattice L with Euclidean function ϕ, is said to be join absorptive (or simply, absorptive) of order t in L; if for each f in L\D, there exists h in D and g in L such that f = hg with 1 ≤ ϕ(g) ≤ t. The main result of the paper states that if I is an ideal of an absorptive subset D of order t in L, then I can be generated by (t+1) or fewer elements. In addition, if I contains an element of ϕ value equals to i+ϕ(I) for some 1 ≤ it with ϕ(I) = inf { ϕ(f) | fI}, then I cannot be a principal ideal in D

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