Necklace ring

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

Definition

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

${\displaystyle \displaystyle c_{n}=\sum _{[i,j]=n}(i,j)a_{i}b_{j}}$

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

See also

• Necklace polynomial

References

• Hazewinkel, Michiel (2009). "Witt vectors. I.". Handbook of Algebra. 6. Elsevier/North-Holland. pp. 319–472. arXiv:0804.3888. ISBN 978-0-444-53257-2. MR 2553661.
• Metropolis, N.; Rota, Gian-Carlo (1983). "Witt vectors and the algebra of necklaces". Advances in Mathematics. 50 (2): 95–125. doi:10.1016/0001-8708(83)90035-X. MR 723197.