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 formula_1 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 formula_1 and formula_3 has components
where formula_5 is the least common multiple of formula_6 and formula_7, and formula_8 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 formula_1 with the power series formula_10.