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}}](../I/m/7e42edaf995ecb49b0ce2392b81fdd3714d9faad.svg)
where [i,j] is the least common multiple of i and j, and (i,j) is their highest common factor.
See also
References
- Hazewinkel, Michiel (2009). "Witt vectors. I.". Handbook of Algebra. 6. Elsevier/North-Holland. pp. 319–472. ISBN 978-0-444-53257-2. MR 2553661. arXiv:0804.3888
. - Metropolis, N.; Rota, Gian-Carlo (1983). "Witt vectors and the algebra of necklaces". Advances in Mathematics. 50 (2): 95–125. MR 723197. doi:10.1016/0001-8708(83)90035-X.
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