Necklace ring

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 $$\textstyle\prod_{n\geq 0} (1{-}t^n)^{-a_n}$$.