Why All Rings Should Have a 1
Name
ring.pdf
Description
Accepted version
Size
258.13 KB
Format
Adobe PDF
Checksum (MD5)
8995e953a6600c7721ff30a8fbd638c4
Author(s)
Poonen, Bjorn
Date Issued
January 2019
Journal
Mathematics Magazine
Publisher
Informa UK Limited
Citation
Poonen, Bjorn et al. "Why All Rings Should Have a 1." Mathematics Magazine 92, 1 (January 2019): 58-62 © 2019 Mathematical Association of America
Version
Original manuscript
Abstract
Should the definition of ring require the existence of a multiplicative identity 1? Emmy Noether, when giving the modern axiomatic definition of a commutativering, in 1921, did not include such an axiom [15, p. 29]. For several
decades, algebra books followed suit [16, x3.1], [18, I.x5]. But starting around 1960, many books by notable researchers began using the term "ring" to mean "ring with 1" [7, 0.(1.0.1)], [14, II.x1], [17, p. XIV], [1, p. 1]. Sometimes a change of heart occurred in a single person, or between editions of a single book, always
towards requiring a 1: compare [11, p. 49] with [13, p. 86], or [2, p. 370] with [3, p. 346], or [4, I.x8.1] with [5, I.x8.1]. Reasons were not given; perhaps it was just becoming increasingly clear that the 1 was needed for many theorems to hold; some good reasons for requiring a 1 are explained in [6]. But is either convention more natural? The purpose of this article is to answer yes, and to give a reason: existence of a 1 is a part of what associativity
should be.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1080/0025570x.2018.1538714
