  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.
