The Big Picture of Commutative Ring For final assignment on my Abstract Algebra class $-$ which is about Commutative Rings with Unity covering roughly Modules, Field of Fractions of an Integral Domain, Integrality and Fields, Prime Ideals, Krull Dimension, Noetherian Integral Domains and Dedekind Domains $-$ I am supposed to write a short essay on the "relationships among the important concepts."
I fully understand and appreciate the wisdom of this assignment, since it forces me to look for the big picture. But being new to higher math, I am still a bonehead when it comes to see the common thread. I would appreciate therefore if you could suggest any tips on relationship among some of them.
Thank you very much for your time and help, while hoping you have had a great Thanksgiving Day last week!
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I am sorry I did not know that it is too broad. Perhaps what it meant by "relationships among the important concepts" is the relationship between any of the above two or three important concepts. Let's make it narrowed down that way. Does it make question answerable? Thank you again.
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I have thought long and hard about this assignment: On one hand the assignment reqiured me to write the relationship between various important concepts, while on the other hand the writing has to be in essay form, meaning that it can't be presented in disjoint, fragmented bullet paragraphs. For these two reasons, I finally settled down by writing a short essay on the history of Commutative Ring, relating various important concepts in a web of history of mathematicians and their influential works that have changed the landscape of Commutative Ring. I have put the writing here below, hoping it benefits somebody someday in the future. Thanks again for all your help.
 A: Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory, and has in turn been applied mainly to these subjects. Due to brevity of this essay, I will focus only on the contribution of algebraic number theory during its nascent years with respect to the emergence of commutative ring theory.
Several of the central areas of number theory, principally Fermat’s Last Theorem and reciprocity laws, were instrumental in the emergence of algebraic number theory. Although the main problems in these areas were originally expressed in terms of integers, it gradually became apparent that the solutions called for embedding the integers in domains of what came to be known as algebraic integers. The following paragraphs look at how those celebrated problems that gave rise to the ideas of algebraic integers.
The famed Swiss mathematician Leonhard Euler (1707 – 1783) in the eighteenth century and several others in the early nineteenth century realized that to prove Fermat’s Last Theorem — the un-solvability in nonzero integers of $x^n + y^n = z^n$ with $ (n > 2)$ — even for small values of $n$, it is necessary to use “complex integers,” which are actually algebraic integers. At another setting, the brilliant German mathematician Carl Friedrich Gauss (1777 – 1855) succeeded in proving the celebrated quadratic reciprocity law, namely that $x^2 \equiv q\pmod p$ is solvable if and only if $x^2 \equiv p \pmod q$ is solvable. This success in turn prodded him to ask further: Is there any reciprocity relation between the solvability of $x^m \equiv q \pmod p$ and $x^m \equiv p \pmod q$ for $m > 2$? Here Gauss eventually concluded, prophetically, that “such a theory demands that the domain of higher arithmetic be endlessly enlarged.”
In the meantime, in the 1840s, a German mathematician Ernst Eduard Kummer (1810–1893) noticed examples of the failure of unique prime factorization in algebraic integers, and he considered it a serious problem. Kummer found “another kind of number” that preserved the property of unique prime factorization, and he called them ideal numbers. Although Kummer’s ideas were brilliant but they were complicated and not clearly formulated, what was needed was a decomposition theory that would apply to more general domains of algebraic integers. This was devised separately by two German mathematicians Richard Dedekind (1831 – 1916) and Leopold Kronecker (1823 – 1891). In Dedekind’s 1871 groundbreaking work, he succeeded in identifying certain subsets that characterized ideal numbers internally, and served as motivation for the introduction of ideals in arbitrary domains of algebraic integers. They are what we now call ring’s ideals.
So in the ﬁrst decade of the twentieth century there were well-established, ﬂourishing, concrete theories of both commutative and non-commutative rings and their ideals, although their roots were mainly in other branches of math such as algebraic number theory. But the time was ripe for the abstract ring concept to emerge. The ﬁrst abstract deﬁnition of a ring was given in 1941 by Abraham  Fraenkel (1891 – 1965), a German-born Israeli mathematician. But it was in the hands of two master algebraists: the German Emmy Noether (1882 – 1935) and Austrian-American Emil Artin (1898 – 1962), that abstract definition of ring was transformed in 1920s into powerful, abstract theories. Noether’s two seminal papers of 1921 and 1927 extended as well as abstracted the decomposition theories of polynomial rings, rings of integers of algebraic number ﬁelds and algebraic function ﬁelds, into abstract commutative rings with the ascending chain condition (ACC) — now called Noetherian rings. 
Also in the same 1927 paper, Noether discussed the Dedekind’s decomposition of ideals as unique products of prime ideals in, respectively, rings of integers of algebraic number ﬁelds and function ﬁelds, in the setting of abstract rings. In particular, she characterized abstract commutative rings in which every nonzero ideal is a unique product of prime ideals. Such rings are now called Dedekind domains. In the meantime Artin, inspired by Noether’s work on commutative rings with the ACC, generalized Wedderburn’s structure theorems on algebras by showing that such rings, with zero radical, can be decomposed into direct sums of simple rings which, in turn, are matrix rings over division rings. Those rings are now called the Artinian rings.
While in Fraenkel’s definition we witness the birth of the abstract ring concept, with Noether and Artin we see the birth of abstract ring theory. Noether and Artin made the abstract ring concept central in algebra by framing in an abstract setting the theorems that were its major inspirations. In this context they introduced and gave prominence to such fundamental algebraic notions as ideal, module, and chain conditions — both ascending and descending. This introduction of module is a watershed, as noted by Michael Atiyah and Ian MacDonald — the celebrated authors of  a book on Commutative Algebra — who wrote that one of the things which distinguishes the modern approach to Commutative Ring is the greater emphasis on modules, rather than just ideals. The extra “elbow room” that this gives makes for greater clarity and simplicity. For instance, an ideal $I$ and its quotient ring $A/I$ are both examples of the modules and so, to a certain extent, can be treated on an equal footing, such that any arguments about ideals or quotient rings can be combined into a single argument about modules.
Ring theory now takes its rightful place, along the now well-established theories of groups and ﬁelds, as one of the pillars of abstract algebra, and the importance of ring theory in algebra and beyond has anything but diminished. As a ﬁnal comment, here is a trivia showcasing the importance of commutative ring : The paper of Richard Taylor and Andrew Wiles, ﬁlling a gap in Wiles’ previously announced proof of Fermat’s Last Theorem after 350 years of waiting, is apparently centered on ring theory as evidenced by its title “Ring-theoretic Properties of Certain Hecke Algebras.”
Sources:
Kleiner, Israel. A History of Abstract Algebra, Birkhauser, 2007; Kline, Morris. Mathematical Thought from Ancient to Modern Times, 3rd volume, Oxford University Press, 1972; Stillwell, John. Mathematics and Its History, 3rd. Ed. Springer, 2010.
A: I would say that commutative algebra is an immediate generalization of the study of common arithmetic and arithmetical polynomials. Just a perspective to start from maybe?
