# Necklace ring

In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983).

## Definition

If A is a commutative ring then the necklace ring over A consists of all infinite sequences (a1,a2,...) 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 (a1,a2,...) and (b1,b2,...) has components

${\displaystyle \displaystyle c_{n}=\sum _{[i,j]=n}(i,j)a_{i}b_{j}}$

where [i,j] is the least common multiple of i and j, and (i,j) is their highest common factor.