Integer-valued polynomials on subsets of upper triangular matrix rings

Abstract

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n ( K ) ( T n ( D ) ) : = { f ∈ T n ( K ) [ x ] | f ( T n ( D ) ) ⊆ T n ( D ) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n ( R 2 ) ( Ω , T n ( R 1 ) ) : = { f ∈ T n ( R 2 ) [ x ] | f ( Ω ) ⊆ T n ( R 1 ) } for some subsets Ω ⊆ T n ( R 1 ) . We generalize Frisch’s result and show that Int T n ( R 2 ) ( T n ( R 1 ) ) : = Int T n ( R 2 ) ( T n ( R 1 ) , T n ( R 1 ) ) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.