In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.
If A is a commutative ring then the necklace ring over A consists of all infinite sequences
(
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
has components where
is the least common multiple of
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
with the power series