Noetherian domains which admit only finitely many star operations

Which Finitely Generated Abelian Groups Admit Equal Growth Functions?

We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsio...

Which Finitely Generated Abelian Groups Admit Isomorphic Cayley Graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.

There are only finitely many Diophantine quintuples

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quint...

ON v–DOMAINS AND STAR OPERATIONS

Let ∗ be a star operation on an integral domain D. Let f(D) be the set of all nonzero finitely generated fractional ideals of D. Call D a ∗–Prüfer (respectively, (∗, v)–Prüfer) domain if (FF−1)∗ = D (respectively, (F vF−1)∗ = D) for all F ∈ f(D). We establish that ∗–Prüfer domains (and (∗, v)–Prüfer domains) for various star operations ∗ span a major portion of the known generalizations of Prüf...

Values of lambda and mu for which there are only finitely many feasible (v,k,lambda,mu)