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I first learned of the concept of "Ideal" when studying Rings in my Algebra course [which was 55 years ago]. I think I have forgotten most of what I learned. However, recently I have encountered the use of the term Ideal in other areas of mathematics that do not involve Rings (or, at least they don't appear to). Also, I have seen references to the "geometry" of ideals which is a mysterious association to me and my meager understanding.

Therefore, I am curious if Ideal as a word represents a kind of concept in math that is common to all of these different areas. Or, better yet, why was Ideal, as a word, used for the particular definitions I have seen in Ring theory.

Physics was my major, not math, but I did take a course in Algebra mostly to become more familiar with notation often used in other areas of math.

Bill Dubuque
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K7PEH
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    I think this question is more appropriate for the History of Mathematics Stack Exchange – lulu Sep 15 '24 at 16:42
  • See https://en.wikipedia.org/wiki/Ideal_(ring_theory)#History – Martin R Sep 15 '24 at 16:43
  • @lulu, historical questions are perfectly appropriate here, and in fact we have almost 3000 of them. – Mikhail Katz Sep 15 '24 at 16:47
  • Please be more specific - in what "other areas of math" have you encountered the term "ideal"? The Ring notion is used more generally in universal algebra (ideal determined varieties) for (equational) algebraic structures whose congruences are determined by a single congruence class. But maybe you are thinking of more general notions of "ideal" (elements), e.g. instances of existential closure or model completions, e.g. Hilbert's method of ideal elements. – Bill Dubuque Sep 15 '24 at 17:40
  • @BillDubuque -- My encountering of other areas where the term "Ideal" is referenced is when I search for Ideal as a means to try and get a better understanding. I mentioned geometry because a non-trivial percentage of my search results had questions about geometry of ideals or some other similar connective words. I brought this up because I had not thought of geometry and the algebra of Ideals in Rings as being connected. Indeed, searching on something like "what is an ideal" in this very Math.SE forum reveals a number of hits where "geometry" is part of the search results. – K7PEH Sep 15 '24 at 19:16
  • "ideal" was coined by Dedekind, as a reification of Kummer's "ideal numbers". These were "ideal" as in "existing in fancy or imagination only". – Arturo Magidin Sep 15 '24 at 19:21
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    Given your prior comment, it seems your are referring to use of ideals in algebraic geometry. Search for "ideal" in the linked wiki article to lean more. – Bill Dubuque Sep 15 '24 at 19:34

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When the absence of unique factorisation in rings of algebraic integers was discovered at the end of the 19th century, new "ideal numbers" were introduced to remedy this. Since these are typically not principal ideals, they are not similar to ideals generated by a specific algebraic integer. Hence the name "ideal" came to be used instead of "number".

Mikhail Katz
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From David Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry (First Edition, Chapter 1, p. 22):

Most significant for modern algebra is surely Dedekind's introduction of ideals of a ring; the name comes from the view that they represent "ideal" (that is to say, "not real") elements of the ring. The search for unique factorization culminated in two major theories, which we shall describe later: Dedekind's unique factorization of ideals into prime ideals in the rings we now call Dedekind domains; and Kronecker's theory of polynomial rings and Lasker's theory of primary decomposition in them.

Dedekind's idea was to represent an element $r\in R$ by the ideal $(r)$ of its multiples; arbitrary ideals might thus be regarded as ideal elements. The ideal $(r)$ determines the element $r$ only up to multiples by units $u$ of R. Since "unique prime factorization" is only unique up to unit multiples anyway, this is just right for generalizing prime factorization. Dedekind sought and found conditions under which a ring has unique factorization of ideals into prime ideals—he showed that this occurs for the ring of all integers in any number field. Dedekind made these definitions, together with the definition of a ring itself, in a famous supplement to later editions (after 1871) of Dirichlet's book on number theory.

Joe
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  • Both answers led me to the same understanding but I selected the other answer by Katz merely because he was first. However, I did decide to buy the book you referenced. – K7PEH Sep 17 '24 at 15:31