I recently learned about irreducible and prime elements in a commutative ring. However, my professor was not quite sure who was the first to make this distinction, or give an example of an irreducible element which is not prime. Was it Dedekind? If so, where did he make this distinction? Could you also give some references regarding the history of commutative rings, UFD, and the evolution of these concepts?


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The phenomenon of nonunique factorization appears to have been first explicitly articulated in the setting of cyclotomic fields, by Kummer in the 1830s and 1840s: $\mathbf Z[\zeta_{23}]$ is the first cyclotomic ring that fails to have unique factorization. One of Kummer's first papers (in 1847) where he discusses how to save unique factorization in cyclotomic rings $\mathbf Z[\zeta_\lambda]$ for prime $\lambda$, using "ideal numbers," is http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002146061&physid=PHYS_0340. (For readers who know algebraic number theory, on the last page of this paper at http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=PPN243919689_0035|LOG_0003&physid=PHYS_0380 Kummer lists some examples of exponents $n$ of class groups of these cyclotomic rings, which are not always equal to the class number $h$ of these rings. For example, when $\lambda$ is $29$ and $41$ he finds $n$ is $2$ and $11$, but $h$ is $8$ and $121$, respectively.) See also Lenstra's article http://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1979a/art.pdf. I do not believe at this time that anyone created special terminology to distinguish primes from irreducibles like we do today.

Moving a few decades later, in Dedekind's famous XI-th Supplement on algebraic number theory in Dirichlet's Lectures on Number Theory (Vorlesungen über Zahlentheorie), which is from the 1870s, there is the example $3 \cdot 7 = (1+2\sqrt{-5})(1-2\sqrt{-5})$ from $\mathbf Z[\sqrt{-5}]$ in Section 159, where Dedekind says each of the four factors is unzerlegbare Zahl, which means an irreducible number. Dedekind's goal was to rescue unique factorization by using ideals instead of elements, and he does not dwell on the phenomenon of non-UFDs long enough to introduce new vocabulary like prime vs. irreducible.

Thanks to Franz Lemmermeyer, I can point you to a reference from 1903 that makes the distinction between primes and irreducibles. On the bottom of page 12 of the book Einleitung in die allgemeine Theorie der algebraischen Groessen, by Gyulya König, (see https://books.google.com/books?id=3LhUAAAAYAAJ&pg=PA12&lpg=PA12&dq=irreduzibel+holoiden&source=bl&ots=aVbTTDuTi6&sig=oOb2Gna_A0T_e8ONlmwsSYy7tEM&hl=en&sa=X&ved=0ahUKEwjGreu8_dLMAhWIQSYKHYlxCY0Q6AEIHDAA#v=onepage&q=irreduzibel%20holoiden&f=false) the terms Prim-größe and irreduzibel are introduced, where the first one means prime element and the second means irreducible irreducible element. At the top of page 13 he writes that each prime element is irreducible but not conversely, and a counterexample is presented over the course of pp. 20-23 in $\mathbf Z[\sqrt{-5}]$, where $2-\sqrt{-5}$ is shown to be irreducible but not prime. That it's not prime follows from $3 \cdot 3 = (2+\sqrt{-5})(2-\sqrt{-5})$. (König was writing before the general concept of a ring was created. He always works in an integral domain, which he calls a Bereich, and he limits his definitions of prime and irreducible to what he calls holoid domains, which are integral domains of characteristic $0$, even though we'd recognize a restriction on the characteristic as an irrelevant hypothesis today. His notation for $\mathbf Z$ is $[1]$, for $\mathbf Q$ is $(1)$, and for $\mathbf Z[\sqrt{d}]$ is $[\sqrt{d}]$.)

Since König is not well-known for his work in algebra, I'll mention another early 20-th century reference: on p. 11 of Krull's 1937 book Idealtheorie (see https://books.google.com/books?id=jCvKBgAAQBAJ&pg=PA11&lpg=PA11&dq=unzerlegbare+element&source=bl&ots=y_Y9BXubOy&sig=ZVQsi9-A-xYKb1TE5UAHE4-p1wk&hl=en&sa=X&ved=0ahUKEwi71Z2NrdHMAhVDVD4KHfjsDUYQ6AEILTAD#v=onepage&q=unzerlegbare%20element&f=false) he defines unzerlegbar and Primelement the way you'd define irreducible and prime, and he also cautions that while every prime element is irreducible, not every irreducible element is prime.

  • $\begingroup$ thanks! looking forward. Did he also coined these terms (in German), or was it later? $\endgroup$
    – David
    May 11, 2016 at 4:21
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    $\begingroup$ @David, I rewrote my answer, so your question ("Did he also...") no longer makes direct sense. The introduction of distinctive terminology goes back at least to 1903 as I point out in my answer. In particular, this rules out Emmy Noether as being a source of the separate terminology, since she was still a student in 1903. $\endgroup$
    – KCd
    May 11, 2016 at 21:44

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