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On Bhargava rings

Mohamed Mahmoud Chems-Eddin, Omar Ouzzaouit, Ali Tamoussit (2023)

Mathematica Bohemica

Let D be an integral domain with the quotient field K , X an indeterminate over K and x an element of D . The Bhargava ring over D at x is defined to be 𝔹 x ( D ) : = { f ∈ K [ X ] : for all a ∈ D , f ( x X + a ) ∈ D [ X ] } . In fact, 𝔹 x ( D ) is a subring of the ring of integer-valued polynomials over D . In this paper, we aim to investigate the behavior of 𝔹 x ( D ) under localization. In particular, we prove that 𝔹 x ( D ) behaves well under localization at prime ideals of D , when D is a locally finite intersection of localizations. We also attempt a classification of integral domains D ...

On delta sets and their realizable subsets in Krull monoids with cyclic class groups

Scott T. Chapman, Felix Gotti, Roberto Pelayo (2014)

Colloquium Mathematicae

Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ(M) ⊆ {1,..., n-2}. Moreover, equality holds for this containment when each class contains a prime divisor from M. In this note, we consider the question of determining...

On prime modules over pullback rings

Shahabaddin Ebrahimi Atani (2004)

Czechoslovak Mathematical Journal

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if R is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime R -modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.

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