Polynomial functions on non-commutative rings - a link between ringsets and null-ideal sets

Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that the integer-valued polynomials on $S$ form a ring, and, on the other hand, sets $S$ such that the set of polynomials in $R[x]$ that are zero on $S$ is an ideal of $R[x]$.Comment: 9 pages, conference paper for "advances in algebra ..." at Ton Duc Thang University, Vietnam, Dec 18-20, 201

Parametrization of Pythagorean triples by a single triple of polynomials

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.Comment: to appear in J. Pure Appl. Algebr

Sylow pp-groups of polynomial permutations on the integers mod pnp^n

We describe the Sylow $p$-groups of the group of polynomial permutations of the integers mod $p^n$

Prime Ideals in Infinite Products of Commutative Rings

In this work we present descriptions of prime ideals and in particular of maximal ideals in products $R = \prod D_\lambda$ of families $(D_\lambda)_{\lambda \in \Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $\prod \mathcal{P}(\max(D_\lambda))$. If every $D_\lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, then this leads to a characterization of the maximal ideals of $R$. If every $D_\lambda$ is a Pr\"ufer domain, we depict all prime ideals of $R$. Moreover, we give an example of a (optionally non-local or local) Pr\"ufer domain such that every non-zero prime ideal is of infinite height

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.Comment: to appear in Monatsh. Math; 15 page

The Antiquity and Evolutionary History of Social Behavior in Bees

A long-standing controversy in bee social evolution concerns whether highly eusocial behavior has evolved once or twice within the corbiculate Apidae. Corbiculate bees include the highly eusocial honey bees and stingless bees, the primitively eusocial bumble bees, and the predominantly solitary or communal orchid bees. Here we use a model-based approach to reconstruct the evolutionary history of eusociality and date the antiquity of eusocial behavior in apid bees, using a recent molecular phylogeny of the Apidae. We conclude that eusociality evolved once in the common ancestor of the corbiculate Apidae, advanced eusociality evolved independently in the honey and stingless bees, and that eusociality was lost in the orchid bees. Fossil-calibrated divergence time estimates reveal that eusociality first evolved at least 87 Mya (78 to 95 Mya) in the corbiculates, much earlier than in other groups of bees with less complex social behavior. These results provide a robust new evolutionary framework for studies of the organization and genetic basis of social behavior in honey bees and their relatives