### Integer-valued polynomials and binomially Noetherian rings

*SHADMAN KAREEM*

^{1}Department of Mathematic, Faculty of Science and Healthy, Koya University. University Park Danielle Mitterrand Boulevard Koya, Erbil, Iraq.

^{2}College of Information Technology and Computer Sciences, Catholic University in Erbil, Erbil, Iraq.

DOI: https://doi.org/10.31972/ticma22.07

**Abstract**

A torsion free as a Z- module ring R with unit is said to be a binomial ring if it is preserved as binomial symbol (a¦i)≔(a(a-1)(a-2)…(a-(i-1)))/i!,

for each a∈R and i ≥ 0. The polynomial ring of integer-valued in rational polynomial Q[X] is defined by Int (Z^X):={h∈Q[X]:h(Z^X)⊂Z} an important example for binomial ring and is non-Noetherian ring. In this paper the algebraic structure of binomial rings has been studied by their properties of binomial ideals. The notion of binomial ideal generated by a given set has been defined. Which allows us to define new class of Noetherian ring using binomial ideals, which we named it binomially Noetherian ring. One of main result the ring Int (Z^({x,y})) over variables x and y present as an example of that kind of class of Noetherian. In general the ring Int(Z^X) over the finite set of variables X and for a particular F subset in Z the rings Int(F^(〖{x〗_1,x_2,…,x_i} ),Z)={h∈Q[x_1,x_2,…,x_i ]:h(F^(〖{x〗_1,x_2,…,x_i} ))⊆ Z} both are presented as examples of that kind of class of Noetherian.

**Key Words:** Binomial ring, Integer-valued polynomial, Binomial ideal, Noetherian rings, binomially Noetherian rings.