### Description

We investigate two kinds of rings of polynomials mapping elements of an algebra to elements of the same algebra.

Both generalize the classical concept of integer-valued polynomials.

An example of the first, commutative, kind of ring is the ring of

polynomials with rational coefficients mapping each n by n integer matrix to an integer matrix. An example of the second, non-commutative, kind of ring is the ring of polynomials with coefficients in the algebra of n by n rational matrices, mapping every n by n integer matrix to an integer matrix.

As with the classical integer-valued polynomials, the main questions concern integral-closure, the Pruefer property, separation of points, polynomial density and non-unique factorization. Both kinds of integer-valued polynomials on algebras require innovative combinations of techniques from commutative ring theory and non-commutative ring

theory for their study. Also, they are useful for constructing rings with certain specified properties (for instance, with respect to factorization).

Both generalize the classical concept of integer-valued polynomials.

An example of the first, commutative, kind of ring is the ring of

polynomials with rational coefficients mapping each n by n integer matrix to an integer matrix. An example of the second, non-commutative, kind of ring is the ring of polynomials with coefficients in the algebra of n by n rational matrices, mapping every n by n integer matrix to an integer matrix.

As with the classical integer-valued polynomials, the main questions concern integral-closure, the Pruefer property, separation of points, polynomial density and non-unique factorization. Both kinds of integer-valued polynomials on algebras require innovative combinations of techniques from commutative ring theory and non-commutative ring

theory for their study. Also, they are useful for constructing rings with certain specified properties (for instance, with respect to factorization).

Status | Active |
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Effective start/end date | 1/05/18 → 30/04/22 |