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Why All Rings Should Have a 1

Author(s)
Poonen, Bjorn
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Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/
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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.
Date issued
2019-01
URI
https://hdl.handle.net/1721.1/126525
Department
Massachusetts Institute of Technology. Department of Mathematics
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
ISSN
0025-570X
1930-0980

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