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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 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 and has components

where is the least common multiple of and , and 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 .

- Hazewinkel, Michiel (2009). "Witt vectors I".
*Handbook of Algebra*. Vol. 6. Elsevier/North-Holland. pp. 319–472. arXiv:0804.3888. Bibcode:2008arXiv0804.3888H. ISBN 978-0-444-53257-2. MR 2553661. - Metropolis, N.; Rota, Gian-Carlo (1983). "Witt vectors and the algebra of necklaces".
*Advances in Mathematics*.**50**(2): 95–125. doi:10.1016/0001-8708(83)90035-X. MR 0723197.