Integer-valued polynomials over block matrix algebras

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials Int(L-s(n)(D)) = {f is an element of L-s(n)(K)[x]vertical bar f(L-s(n)(D))subset of L-s(n)(D)} for each 1 <= s <= n - 1, where D is an integral domain with quotient field K and L-s(n) (D) is a block matrix ring between upper triangular matrix ring T-n(D) and full matrix ring M-n(D). In fact, we have L-n-1(n)(D) = T-n(D). it is known that the sets of integer-valued polynomials Int(T-n(D)) = {f is an element of T-n (K) [x]vertical bar f (T-n(D)) subset of T-n(D)} and Int(M-n(D)) = {f is an element of M-n(K)[x]vertical bar f(M-n(D)) subset of M-n(D)} are rings. We state some relations between the rings Int(T-s(D)), Int(Mn-s(D)) and the partitions of Int (L-s(n)(D)). Then, we show that, the set Int(L-s(n)(D)) is a ring for each 1 <= s <= n - 1. Further, it is proved that if the ring Int(Mn-s(D)) is not Noetherian then the ring Int(L-s(n)(D)) is not Noetherian, too. Finally, some properties and relations are stated between the rings Int(L-s(n)(D)), Int(T-s(D)) and Int(Mn-s(D)).