### Abstract

In this article, we study the ring of integer-valued polynomials over some matrix rings. Let D be an integral domain with the field of fractions K and I be an ideal of D. We introduce Int(M_{n}(I), M_{n}(D)) = {f ∈ M_{n}(K)[x] | f(M_{n}(I)) ⊆ M_{n}(D)}, then we prove that (M_{n}(I), M_{n}(D) is a ring. We show that Int(M_{n}(I), M_{n}(Z) is not Noetherian where I is an ideal of Z. We present a direct proof to show that the set Int(T_{n}(E); T_{n}(D)) = {f ∈ T_{n} (K)[x] | f(T_{n}(E)) ⊆ T_{n}(D)} is a ring where T_{n}(D) is the upper triangular matrix ring over D and E is a subset of D containing 0. We find out that, if Int(E,D) is not Noetherian then Int(T_{n}(E), T_{n}(D)) is a non-Noetherian ring. Also, we state a lower bound for Krull dimension of integer-valued polynomials ring Int(T_{n}(E), T_{n}(D)). We further introduce Int(H_{n}(I), H_{n}(D)) = {f ∈ H_{n}(K)[x] | f (H_{n}(I)) ⊆ H_{n}(D)}, where H_{n}(D) = f[a_{ij}] ∈ T_{n}(D) | a_{11} = a_{22} = … = a_{nn}} and we demonstrate that it is a ring. Finally, we illustrate that if Int(H_{n}(E), H_{n}(D)) is a Noetherian ring then Int(E,D) is Noetherian, where E is a subset of D containing 0.

Original language | English |
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Pages (from-to) | 1077-1090 |

Number of pages | 14 |

Journal | Communications in Algebra |

Volume | 47 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 |

### Keywords

- Integer-valued polynomial
- matrix ring

## Cite this

*Communications in Algebra*,

*47*(3), 1077-1090. https://doi.org/10.1080/00927872.2018.1499926