# Integer-valued polynomials over subsets of matrix rings

J. Sedighi Hafshejani, A. R. Naghipour, A. Sakzad

Research output: Contribution to journalArticleResearchpeer-review

### 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(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.

Original language English 1077-1090 14 Communications in Algebra 47 3 https://doi.org/10.1080/00927872.2018.1499926 Published - 2019

### Keywords

• Integer-valued polynomial
• matrix ring

### Cite this

Sedighi Hafshejani, J. ; Naghipour, A. R. ; Sakzad, A. / Integer-valued polynomials over subsets of matrix rings. In: Communications in Algebra. 2019 ; Vol. 47, No. 3. pp. 1077-1090.
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title = "Integer-valued polynomials over subsets of matrix rings",
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(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.",
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author = "{Sedighi Hafshejani}, J. and Naghipour, {A. R.} and A. Sakzad",
year = "2019",
doi = "10.1080/00927872.2018.1499926",
language = "English",
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pages = "1077--1090",
journal = "Communications in Algebra",
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Integer-valued polynomials over subsets of matrix rings. / Sedighi Hafshejani, J.; Naghipour, A. R.; Sakzad, A.

In: Communications in Algebra, Vol. 47, No. 3, 2019, p. 1077-1090.

Research output: Contribution to journalArticleResearchpeer-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

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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 -