Titlebook: Abstract Algebra; Suitable for Self-St Marco Hien Textbook 2024 The Editor(s) (if applicable) and The Author(s), under exclusive license to

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Rings and Ideals,ill focus on their basic properties in this chapter. In particular, we will discuss the concept of an ideal, which leads to factor rings analogous to factor groups. More specific properties will be discussed in the following chapters.

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Euclidean Rings, Principal Ideal Rings, Noetherian Rings,th remainder, there is a unique prime factor decomposition, there is a greatest common divisor, . Similar properties can be found in the ring .[.] for a field .. In the next two chapters, we will examine such (wishfull) properties that rings can have, abstractly – and also see examples of rings that do not posess these properties.

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Irreducible Polynomials in UFDs,e further investigation of the roots of . in Galois theory. In this chapter, we develop techniques that can help us whenever the field . is the quotient field of a UFD .. For example for ., this is the case with ..

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