I have encountered various abstract algebra resources that prove the impossibility of number systems with plural additive inverses for a given element, generally through the substitution property of equality. Historically, did these proofs arise in response to ill-conceived number systems with multiple additive inverses or are they more so used to illustrate limitations on the properties of certain algebraic structures?

  • $\begingroup$ I think these elementary exercises are just there to give the student practice with the terminology and what it means. It is sort of curious that, while rings can have noncommutative multiplication, commutativity of addition is actual forced by the other ring axioms. See math.stackexchange.com/questions/346375/… $\endgroup$ – KCd Jan 6 at 20:19
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    $\begingroup$ Uniqueness of the inverse follows from just the group axioms. Since rings are defined to be commutative groups with respect to addition the question should be who first proved uniqueness of the inverse in groups. Hölder mentions that it follows in Zurückfühnmg einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen (1889), he uses the cancelation law as one of the axioms, see Wussing, The Genesis of the Abstract Group Concept, p.245. $\endgroup$ – Conifold Jan 7 at 22:57
  • $\begingroup$ @Conifold Thank you for the citation. $\endgroup$ – bblohowiak Jan 7 at 23:21
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    $\begingroup$ In the early 20th century, lots of parts of algebra were axiomatized. In connection with that, there would be efforts to see whether the axioms were independent or redundant. $\endgroup$ – Gerald Edgar Jan 8 at 1:23

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