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Rings and Algebras

  • Guerino Mazzola
Chapter
Part of the Computational Music Science book series (CMS)

Abstract

A (unitary) ring is a triple (R, α, μ) where (R, α) is an abelian group whose operation α is written additively (α(r, s) = r + s) with neutral element 0 R , and (R, μ) is monoid, written multiplicatively \( ({\mu}(r, s) = {r}\cdot{s})\) with multiplicative neutral element 1 R such that these operations are coupled by distributivity, i.e., \( (r + s) \cdot {t} = {r} \cdot {t}+{s} \cdot {t}, {t} \cdot (r + s) = {t} \cdot {r} + {t} \cdot {s}\) for all \( {r, s, t} \ {\in} \ {R}\).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Guerino Mazzola
    • 1
  1. 1.School of MusicUniversity of MinnesotaMinneapolisUSA

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