Why All Rings Should Have a 1
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 starti...
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| Format: | Article |
| Language: | English |
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Informa UK Limited
2020
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| Online Access: | https://hdl.handle.net/1721.1/126525 |
| _version_ | 1826207437283131392 |
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| author | Poonen, Bjorn |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Poonen, Bjorn |
| author_sort | Poonen, Bjorn |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T13:49:39Z |
| format | Article |
| id | mit-1721.1/126525 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:49:39Z |
| publishDate | 2020 |
| publisher | Informa UK Limited |
| record_format | dspace |
| spelling | mit-1721.1/1265252022-09-28T16:27:03Z Why All Rings Should Have a 1 Poonen, Bjorn Massachusetts Institute of Technology. Department of Mathematics 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. National Science Foundation (Grants DMS-1069236, DMS-1601946) Simons Foundation (Grants 340694, 402472) 2020-08-07T20:45:40Z 2020-08-07T20:45:40Z 2019-01 2016-06 2019-11-18T18:01:27Z Article http://purl.org/eprint/type/JournalArticle 0025-570X 1930-0980 https://hdl.handle.net/1721.1/126525 Poonen, Bjorn et al. "Why All Rings Should Have a 1." Mathematics Magazine 92, 1 (January 2019): 58-62 © 2019 Mathematical Association of America en http://dx.doi.org/10.1080/0025570x.2018.1538714 Mathematics Magazine Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Informa UK Limited MIT web domain |
| spellingShingle | Poonen, Bjorn Why All Rings Should Have a 1 |
| title | Why All Rings Should Have a 1 |
| title_full | Why All Rings Should Have a 1 |
| title_fullStr | Why All Rings Should Have a 1 |
| title_full_unstemmed | Why All Rings Should Have a 1 |
| title_short | Why All Rings Should Have a 1 |
| title_sort | why all rings should have a 1 |
| url | https://hdl.handle.net/1721.1/126525 |
| work_keys_str_mv | AT poonenbjorn whyallringsshouldhavea1 |