Divisibility in Integral Domains

Dinesh Khattar () and Neha Agrawal ()
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Dinesh Khattar: University of Delhi, Department of Mathematics
Neha Agrawal: University of Delhi, Department of Mathematics

Chapter Chapter 7 in Ring Theory, 2023, pp 245-289 from Springer

Abstract: Abstract In the previous chapter, we defined the notion of divisibility in the integral domain F[x], and then proceeded to discuss the irreducibility of a polynomial. Having laid down this groundwork, we will now build upon it by extending the concept of divisibility to arbitrary integral domains. Further harnessing this concept, we shall define special type of elements – irreducible and prime elements, similar to prime elements in $$\mathbb{Z}$$ Z or irreducible polynomials in F[x]. The second story of our construction will contain two important classes of integral domains, namely, the unique factorization domains and the Euclidean domains, along with their distinctive properties. The definition of the unique factorization domain arises as an application of the fundamental theorem of arithmetic, which is true in the ring of integers, to more abstract rings.

Date: 2023
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DOI: 10.1007/978-3-031-29440-2_7

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