Necklace ring

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In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983).


If A is a commutative ring then the necklace ring over A consists of all infinite sequences (a1,a2,...) of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of (a1,a2,...) and (b1,b2,...) has components

where [i,j] is the least common multiple of i and j, and (i,j) is their highest common factor.

See also[edit]


  • Hazewinkel, Michiel (2009). "Witt vectors. I.". Handbook of Algebra. 6. Elsevier/North-Holland. pp. 319–472. arXiv:0804.3888. Bibcode:2008arXiv0804.3888H. ISBN 978-0-444-53257-2. MR 2553661.
  • Metropolis, N.; Rota, Gian-Carlo (1983). "Witt vectors and the algebra of necklaces" (PDF). Advances in Mathematics. 50 (2): 95–125. doi:10.1016/0001-8708(83)90035-X. MR 0723197.