Necklace ring

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

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