# 4: Ideals and Homomorphisms and test

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The first three chapters of this text tell the story of unique factorization. The culmination is the result that any Euclidean domain is a unique factorization domain; that is, in an integral domain with a well-behaved division algorithm, a nonzero nonunit necessarily factors uniquely into irreducibles. In order to expediently develop that result, we ignored many concepts that are otherwise interesting and useful in a first course in abstract algebra. This chapter is a coda that seeks to fill in some of those gaps.

In Section 4.1, we expand on the definition of ideal introduced in Section 2.4 and explore non-principal ideals. No math course is complete without a discussion of functions of some sort; we explore homomorphisms in Section 4.2 Finally, in Section 4.3, we introduce prime and maximal ideals, as well as the notion of congruence modulo II and use ideals to build new rings from old. We conclude with an exploration and proof of the First Isomorphism Theorem.

• 4.1: Ideals in general
In this section, we explore ways of describing non-principal ideals. We also explore properties of ideals, as well as their connections to other fields of mathematics.
• 4.2: Homomorphisms
This section assumes a familiarity with the idea of function from a set-theoretic point of view, as well as the concepts of injective (one-to-one), surjective (onto), and bijective functions (one-to-one correspondences).
• 4.3: Quotient Rings: New Rings from Old
In this section, we explore a way of building new rings from old by means of ideals. To better understand these new rings, we will also define two new classes of ideals: prime ideals, and maximal ideals. We end by briefly connecting these rings to a familiar problem from high school algebra.

This page titled 4: Ideals and Homomorphisms and test is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael Janssen & Melissa Lindsey via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.