### 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 |
---|---|

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

}

*Communications in Algebra*, vol. 47, no. 3, pp. 1077-1090. https://doi.org/10.1080/00927872.2018.1499926

**Integer-valued polynomials over subsets of matrix rings.** / Sedighi Hafshejani, J.; Naghipour, A. R.; Sakzad, A.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Integer-valued polynomials over subsets of matrix rings

AU - Sedighi Hafshejani, J.

AU - Naghipour, A. R.

AU - Sakzad, A.

PY - 2019

Y1 - 2019

N2 - 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(Mn(I), Mn(D)) = {f ∈ Mn(K)[x] | f(Mn(I)) ⊆ Mn(D)}, then we prove that (Mn(I), Mn(D) is a ring. We show that Int(Mn(I), Mn(Z) is not Noetherian where I is an ideal of Z. We present a direct proof to show that the set Int(Tn(E); Tn(D)) = {f ∈ Tn (K)[x] | f(Tn(E)) ⊆ Tn(D)} is a ring where Tn(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(Tn(E), Tn(D)) is a non-Noetherian ring. Also, we state a lower bound for Krull dimension of integer-valued polynomials ring Int(Tn(E), Tn(D)). We further introduce Int(Hn(I), Hn(D)) = {f ∈ Hn(K)[x] | f (Hn(I)) ⊆ Hn(D)}, where Hn(D) = f[aij] ∈ Tn(D) | a11 = a22 = … = ann} and we demonstrate that it is a ring. Finally, we illustrate that if Int(Hn(E), Hn(D)) is a Noetherian ring then Int(E,D) is Noetherian, where E is a subset of D containing 0.

AB - 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(Mn(I), Mn(D)) = {f ∈ Mn(K)[x] | f(Mn(I)) ⊆ Mn(D)}, then we prove that (Mn(I), Mn(D) is a ring. We show that Int(Mn(I), Mn(Z) is not Noetherian where I is an ideal of Z. We present a direct proof to show that the set Int(Tn(E); Tn(D)) = {f ∈ Tn (K)[x] | f(Tn(E)) ⊆ Tn(D)} is a ring where Tn(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(Tn(E), Tn(D)) is a non-Noetherian ring. Also, we state a lower bound for Krull dimension of integer-valued polynomials ring Int(Tn(E), Tn(D)). We further introduce Int(Hn(I), Hn(D)) = {f ∈ Hn(K)[x] | f (Hn(I)) ⊆ Hn(D)}, where Hn(D) = f[aij] ∈ Tn(D) | a11 = a22 = … = ann} and we demonstrate that it is a ring. Finally, we illustrate that if Int(Hn(E), Hn(D)) is a Noetherian ring then Int(E,D) is Noetherian, where E is a subset of D containing 0.

KW - Integer-valued polynomial

KW - matrix ring

UR - http://www.scopus.com/inward/record.url?scp=85057315478&partnerID=8YFLogxK

U2 - 10.1080/00927872.2018.1499926

DO - 10.1080/00927872.2018.1499926

M3 - Article

VL - 47

SP - 1077

EP - 1090

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 3

ER -