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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
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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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