Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis∗ 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) en- suring uniqueness of the quotient and remainder in the Division Algorithm; (2) permit- ting unique base expansion with respect to any nonzero nonunit in the ring; (3) allowing explicit solutions to B´ezout'sidentity with norm constraints. The two most popular examples of Euclidean domains, the ring of integers Z and the ring F[ξ] of polynomials over a field F, possess slightly different properties. For example, in Z the quotient and remainder from the Division Algorithm are generally not unique (becoming so when one restricts to positive integers), while the quotient and remainder in F[ξ] are unique. Indeed, Jodeit Jr. [1967] shows that any Euclidean domain with a unique Division Algorithm is isomorphic to either a field or to the polynomial ring over a field. The differences in the two rings Z and F[ξ] also shows up in two other commonly presented results which derive from the Division Algorithm: (1) the expansion of elements of the ring in terms of a base (which is taken to be a nonzero nonunit); (2) the computation, using the Euclidean Algorithm, of solutions to B´ezout'sidentity for coprime ring elements, and with constraints on the Euclidean norms of the solution. For the base expansion in Z, to ensure uniqueness one again needs to restrict to positive integers, whereas the base expansion is always unique in F[ξ]. Moreover, the proofs in the two cases are typically carried out separately, or the proof of one is (not entirely accurately) suggested to follow just like the proof of the other. This leads to the natural question, \Is there a property of subsets of Euclidean domains which ensures, in these subsets: (1) uniqueness of the quotient and remainder; (2) uniqueness of base expansion; (3) norm bounds in the Euclidean Algorithm." We show that there is indeed such a property, and it is quite simple|we call this property \δ-positivity." Let us review the basic features of Euclidean domains, and provide the new definitions that will be used to prove some useful results for Euclidean domains having these properties. If A is a subset of B we write A ⊆ B, using the notation A ⊂ B to denote proper inclusion. We denote by Z>0 the set of positive integers and by Z≥0 the set of nonnegative integers. For an integral domain R we let 1R denote the unit element and 0R denote the zero element. For a field F, F[ξ] denotes the polynomial ring with coefficients in F. By deg(A) we denote the degree of A 2 F[ξ], with the convention that deg(0F[ξ]) = −∞. Since there is not perfect agreement on what properties should be assigned to a Euclidean norm, let us first say exactly what we mean in this paper by a Euclidean domain. A ∗Professor, Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada Email: [email protected], URL: http://www.mast.queensu.ca/~andrew/ Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada 1 2 A. D. Lewis Euclidean domain is a pair (R; δ), where R is an integral domain and where the map δ : R ! Z≥0, called the norm, has the following properties: 1. if a; b 2 R and if ab 6= 0R, then δ(ab) ≥ δ(a); 2. if a; b 2 R with b 6= 0R, then there exists q; r 2 R such that (a) a = qb + r and such that (b) δ(r) < δ(b). We shall make use of the following facts about Euclidean domains, without explicit mention: 1. δ(0R) < δ(1R); 2. if a 2 R n f0Rg then δ(a) ≥ δ(1R); 3. δ(a) = δ(0R) if and only if a = 0R; 4. for a 2 R n f0Rg, δ(ab) = δ(a) if and only if b is a unit; 5. a 2 R is a unit if and only if δ(a) = δ(1R). Now let us give a few new definitions. 1 Definition: Let (R; δ) be a Euclidean domain. (i) A subset C ⊆ R is trivial if C = f0Rg, and is nontrivial otherwise. (ii) A nonempty subset C ⊆ R is δ-closed if, for each a; b 2 C with b 6= 0R, there exists q; r 2 C such that a = qb + r and such that δ(r) < δ(b). (iii) A subset C ⊆ R admits a unique Division Algorithm if, for each a; b 2 C with b 6= 0R, there exists unique q; r 2 C such that a = qb + r and such that δ(r) < δ(b). (iv) A nonempty subset P ⊆ R is δ-positive if, for each a; b 2 P , we have δ(a − b) ≤ maxfδ(a); δ(b)g. • In this paper we will be interested in nontrivial, δ-closed, and δ-positive subsemirings of Euclidean domains, recalling that a subsemiring S ⊆ R has the property that if a; b 2 S then ab 2 S and a + b 2 S. Let us give the two primary examples which illustrate the preceding concepts. 2 Examples: 1. For the ring Z we take the usual Euclidean norm: δ(k) = jkj. One can easily verify using elementary properties of integers that the subset Z≥0 ⊆ Z is a nontrivial, δ-closed, and δ-positive subsemiring. Note, however, that Z is not a δ- positive subset of itself since, for example, δ(1 − (−2)) = 3 > maxfδ(1); δ(2)g. 2. Let F be a field and define δ : F[ξ] ! Z≥0 by ( 0;A = 0 ; δ(A) = F[ξ] deg(A) + 1;A 6= 0F[ξ]: Subsets of Euclidean domains possessing a unique division algorithm 3 The pair (F[ξ]; δ) is then well known to be a Euclidean domain. We claim that F[ξ] is a δ-positive subset of itself. If either A or B is nonzero, then δ(A − B) = deg(A − B) + 1 ≤ maxfdeg(A); deg(B)g + 1 = maxfdeg(A) + 1; deg(B) + 1g = maxfδ(A); δ(B)g; and, if A = B = 0F[ξ], then δ(A−B) = maxfδ(A); δ(B)g. This shows that F[ξ] is indeed δ-positive. • The following property of nontrivial δ-closed subsets is useful. 3 Lemma: If (R; δ) is a Euclidean domain and if S ⊆ R is a nontrivial δ-closed subsemir- ing, then 0R; 1R 2 S. Proof: Let b 2 S − f0Rg. Since S is δ-closed there exists q; r 2 S such that b = qb + r with δ(r) < δ(b). We claim that this implies that q = 1R and r = 0R. Suppose that q 6= 1R. Then δ(b) ≤ δ((1R − q)b) = δ(r) < δ(b) which is a contradiction. Thus q = 1R, and it then follows that r = 0R. Let us first explore the relationship between δ-positivity and uniqueness in the Division Algorithm. Note that, for the Euclidean domain (Z; δ), we do not generally have such uniqueness since, for example, we can write 6 = 1 · 5 + 1 = 2 · 5 − 4. 4 Proposition: If (R; δ) is a Euclidean domain and if S is a nontrivial, δ-closed, and δ- positive subsemiring of R, then S admits a unique Division Algorithm. Proof: Suppose that a = q1b + r1 = q2b + r2 for q1; q2; r1; r2 2 S with δ(r1); δ(r2) < δ(b). Then (q1 − q2)b = r2 − r1, and so δ((q1 − q2)b) = δ(r1 − r2) ≤ maxfδ(r1); δ(r2)g < δ(b); using δ-closedness of S. This implies that (q1 − q2)b = 0R. Since b 6= 0R this implies that q1 − q2 = 0R and so q1 = q2. We then immediately have r1 = r2. The condition of δ-positivity is, in certain circumstances, also necessary for uniqueness in the Division Algorithm. 5 Proposition: Let (R; δ) be a Euclidean domain and let S ⊆ R be a nontrivial, δ-closed subsemiring with the following properties: (i) S generates R as a ring; (ii) S admits a unique Division Algorithm. Then S is δ-positive. Proof: Note that since S is a subsemiring, S generates R as a ring if and only if, for every r 2 R, it holds that either r 2 S or −r 2 S. Suppose that S is not δ-positive so that δ(a − b) > maxfδ(a); δ(b)g for some a; b 2 S. Suppose that b − a 2 S. Then b = 0R · (b − a) + b; δ(b) < δ(b − a); b = 1R · (b − a) + a; δ(a) < δ(b − a); which shows that S does not admit a unique Division Algorithm. An entirely similar argument gives the same conclusion when a − b 2 S. 4 A. D. Lewis Next we show that base expansion is valid in δ-positive subsets. Again, while base expansions exist for all integers, in order to ensure uniqueness of the coefficients in the expansion, one needs to restrict to positive integers to obtain uniqueness. Much of the proof we give is to be found in standard texts, but we give all of the details in order to illustrate exactly where our additional hypothesis of δ-positivity is used. 6 Proposition: Let (R; δ) be a Euclidean domain, let S ⊆ R be a nontrivial, δ-closed, and δ-positive subsemiring, and let b 2 S be a nonzero nonunit. Then, given a 2 S n f0Rg, there exists a unique k 2 Z≥0 and unique r0; r1; : : : ; rk 2 S such that (i) rk 6= 0R, (ii) δ(r0); δ(r1); : : : ; δ(rk) < δ(b) and 2 k (iii) a = r0 + r1b + r2b + ··· + rkb .