## Integer-valued polynomials over matrix rings.(English)Zbl 1272.16028

Let $$D$$ be a commutative integral domain with field of fractions $$K$$ and let $$M_n(R)$$ be the ring of $$n\times n$$ matrices with entries from the ring $$R$$. Then $\text{Int}(M_n(D))=\{f\in M_n(K)[x]\mid f(M_n(D))\subseteq M_n(D)\}$ is a ring and this ring is called a ring of integer-valued polynomials over $$M_n(D)$$. The purpose of this article is to investigate the properties of $$\text{Int}(M_n(D))$$ and its ideals. In particular, if $$\mathbb Z$$ is the ring of the integers, then the author proves that $$\text{Int}(M_n(\mathbb Z))$$ is a non-Noetherian ring.

### MSC:

 16S36 Ordinary and skew polynomial rings and semigroup rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 16S50 Endomorphism rings; matrix rings

### Keywords:

integer-valued polynomials; matrices; non-commutative rings
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### References:

 [1] Cahen P.-J., Integer-Valued Polynomials (1997) [2] Frisch , S. ( 2005 ).Polynomial Separation of Points in Algebras. Arithmetical Properties of Commutative Rings and Monoids.Lect. Notes Pure Appl. Math. 241 . · Zbl 1092.13027 [3] DOI: 10.1090/conm/499/09805 [4] DOI: 10.1016/j.jalgebra.2010.06.024 · Zbl 1219.16027 [5] DOI: 10.1080/00927872.2011.571732 · Zbl 1246.13038
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