In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.
Definition[edit]
If A is a commutative ring then the necklace ring over A consists of all infinite sequences 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 and has components
![{\displaystyle \displaystyle c_{n}=\sum _{[i,j]=n}(i,j)a_{i}b_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e42edaf995ecb49b0ce2392b81fdd3714d9faad)
where is the least common multiple of and , and is their greatest common divisor.
![{\displaystyle [i,j]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fff46b0cabba9dd4a48696b0d6f85eb38192a021)
This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence with the power series .
See also[edit]
References[edit]
- Hazewinkel, Michiel (2009). "Witt vectors I". Handbook of Algebra. Vol. 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". Advances in Mathematics. 50 (2): 95–125. doi:10.1016/0001-8708(83)90035-X. MR 0723197.