1. Definition

2. See also

3. References

In mathematics, the necklace ring is a ring introduced by {{harvs|txt|last=Metropolis|last2=Rota|year=1983}} to elucidated the multiplicative properties of necklace polynomials.

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 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 (a₁,a₂,...) with the power series .

See also

• Witt vector

References

• {{cite book|mr=2553661 | last1=Hazewinkel | first1=Michiel | author-link = Michiel Hazewinkel |chapter=Witt vectors I |title= Handbook of Algebra |volume=6 |pages=319–472 |publisher=Elsevier/North-Holland |year= 2009 |arxiv=0804.3888 |isbn=978-0-444-53257-2|bibcode=2008arXiv0804.3888H }}
• {{Cite journal |last1=Metropolis |first1=N. |author1-link = Nicholas Metropolis |last2=Rota |first2=Gian-Carlo |author2-link=Gian-Carlo Rota |title=Witt vectors and the algebra of necklaces |doi=10.1016/0001-8708(83)90035-X |mr=723197 |year=1983 |journal=Advances in Mathematics |volume=50 |issue=2 |pages=95–125 | url = https://www.sciencedirect.com/science/article/pii/000187088390035X/pdf?md5=3cfd1a39f5235b827cb18f8a3e99ccc2&pid=1-s2.0-000187088390035X-main.pdf}}

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