Necklace ring explained

In mathematics, the necklace ring is a ring introduced by to elucidate the multiplicative properties of necklace polynomials.

Definition

If A is a commutative ring then the necklace ring over A consists of all infinite sequences

(a _{1,} a _{2,} ...) 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 (a _{1,} a _{2,} ...) and (b _{1,} b _{2,} ...) has components\displaystyle c _{n=\sum} _{[i,j]=n} (i,j)a _{ib} _{j}

where

[i,j] is the least common multiple of i and j , and (i,j) is their greatest common divisor.

This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence

(a _{1,} a _{2,} ...) with the power series style\prod _{n\geq} ( 1 { - }t ^{n)} ^{- a _{n}} .

See also

• Witt vector

References

• Book: 2553661 . Hazewinkel . Michiel . Michiel Hazewinkel . Witt vectors I . Handbook of Algebra . 6 . 319–472 . . 2009 . 0804.3888 . 978-0-444-53257-2. 2008arXiv0804.3888H.
• Metropolis . N. . Nicholas Metropolis . Rota . Gian-Carlo . Gian-Carlo Rota . Witt vectors and the algebra of necklaces . 10.1016/0001-8708(83)90035-X . 723197 . 1983 . . 50 . 2 . 95–125 . free .