A Remark on Sums of Units

Bernhard Herwig and Martin Ziegler

March 22, 2001

Dedicated to Rüdiger Göbel on the occasion of his 60th birthday.

Abstract For every n ≥ 2 we construct a factorial domain R for which n minimal with the property that every element can be written as the sum of at most n units.

For any ring R let u(R) be the smallest number n such that every element can be written as the sum of at most n units. If no such n exists, set u(R) = ∞.

Peter Vamos computed in [Va] u(R) for various rings and found examples with values 1, 2, 3, ∞. We will show that all finite values occur for factorial domains. For a slightly different definition of unit sum number see [GPS].

Everything will follow from the following proposition.

Proposition 1 Let R be an integral domain, a a non–zero element of R and n a natural number ≥ 2. Then R is contained in a domain R′ with the following properties

1. a is the sum of n units in R′.

2. If an element of R is the sum of k < n units in R′, it is the sum of k units in R.

We call a domain with property 2. an n–extension of R.

Proof: Consider the polynomial ring P = R[x_{1}, …, x_{n−1}]. Let S be the multiplicative monoid generated by x_{1}, …, x_{n−1} and w = −x_{1} − ⋯ − x_{n−1} + a. R′ will be the quotient ring P_{S}. Clearly, a is a sum of n units in R′:

a = x_{1} + ⋯ + x_{n−1} + w.

Now assume that r ∈ R is a sum of k < n units in R′. The units in R′ have the form ust^{−1} for an R–unit u and elements s, t of S (because of the special form of the generators of S). Hence for R–units u_{1}, …, u_{k} and elements s_{0}, …, s_{k} of S we have

Mathematics Subject Classification(2000) 13B30, 16U60

rs_{0} − u_{1}s_{1} − ⋯ − u_{k}s_{k} = 0.

We write s_{i} = μ_{i}w^{m_{i}} for monomials μ_{i}. If we denote by f the polynomial

f(x_{1}, …, x_{n−1}, x_{n}) = rμ_{0}x_{n}^{m_{0}} − u_{1}μ_{1}x_{n}^{m_{1}} − ⋯ − u_{k}μ_{k}x_{n}^{m_{k}}

from R[x_{0}, …, x_{n−1}, x_{n}], we have f(x_{1}, …, x_{n−1}, w) = 0 and it follows that f is a multiple of x_{n} − w = x_{1} + … + x_{n} − a. On the other hand, f contains at most n monomials. Hence, by the next lemma, f = 0 and we have

f(1, …, 1) = r − u_{1} − ⋯ − u_{k} = 0.

Thus r is the sum of k units in R.

Lemma 2 Let R be an integral domain and a a non–zero element of R. If the polynomial f ∈ R[x_{1}, …, x_{n}] is non–zero and divisible by x_{1} + ⋯ + x_{n} − a, it contains more than n monomials.

Proof: Write f = g · (x_{1} + ⋯ + x_{n} − a). Let m be the total degree of g. For each i let μ_{i} a monomial from g which has total degree m and maximal degree in x_{i}. Then the monomials μ_{1}x_{1}, …, μ_{n}x_{m} all occur in f. On the other hand all monomials of g which have minimal total degree survive (multiplied by a) in f. If follows that f contains at least n + 1 monomials.

Theorem 3 Each integral domain R has an n–extension R′ with u(R′) ≤ n.

Proof: Choose a well ordering (a_{α}) of the elements of R. We construct an ascending chain of domains R_{α} starting with R_{0} = R. We choose R_{α+1} by the proposition as an n–extension of R_{α} where a_{α} becomes a sum of n units. For limit ordinals we take R_{λ} to be the union of the earlier R_{α}. The union of this chain is an n-extension of R in which every element of R is a sum of n units. If we iterate this process countably many times and take the union of the resulting chain of extensions we find the desired R′.

Corollary 4 For each n ≥ 2 there is a factorial domain R with u(R) = n

Proof: If we apply the theorem to the ring of integers to obtain an n–extension R with u(R) ≤ n. Since the integer n is not the sum of fewer than n units, we have u(R) = n. If we rewind the proof, we see that R is a quotient ring of the polynomial ring over the integers with infinitely many variables, hence factorial.

References

[Va] P.Vamos, 2–good rings, talk at the Festkolloquium and Conference on Algebra, Model Theory and Theoretical Physics celebrating Rüdiger Göbel’s 60th Birthday in Essen, February 2001.

[GPS] B.Goldsmith, S.Pabst, A.Scott: Unit Sum Numbers of Rings and Modules, in: Quart. Journ. Math. Oxford (2), 49 (1998), 331-344.