DIVISION CHAINS AND QUASI-EUCLIDEAN RINGS

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1 DIVISION CHAINS AND QUASI-EUCLIDEAN RINGS D.D. ANDERSON and J.R. JUETT Communicated by Alexandru Zaharescu Let R be a commutative ring with identity. For a, b R and n 1, an n- stage division chain is a sequence of equations a = bq 1 + r 1, b = r 1q 2 + r 2,..., r n 2 = r n 1q n + r n. If r n = 0, we say that b is an n-stage divisor of a. The ring R is n-quasi-euclidean (resp., quasi-euclidean) if for nonzero a, b R, b is an n- stage divisor of a (resp., b is an m-stage divisor of a for some m 1 depending on a and b). We study n-quasi-euclidean rings and quasi-euclidean rings and their relation to matrix reduction. We give a number of characterizations of quasi- Euclidean rings. Now, R is 2-quasi-Euclidean, precisely when it is B ezout with stable rank 1, or, equivalently, for a, b R there is an x R with (a, b) = (a+bx). We show that R is 3-quasi-Euclidean if for a, b R there exist c, d R with (a, b) = (ca + (1 cd)b) and give a similar characterization of n-quasi-euclidean rings. We characterize when the monoid ring R[X; S] or power series ring R[[X]] is quasi-euclidean. AMS 2010 Subject Classication: 13A05, 13F05, 13F07, 13J05, 16S36. Key words: B ezout ring, division chain, elementary divisor ring, Euclidean ring, Hermite ring, quasi-euclidean ring, stable rank. 1. INTRODUCTION Let R be a commutative ring with identity. For a, b R and n 1, an n-stage division chain is a sequence of equations a = bq 1 + r 1, b = r 1 q 2 + r 2,..., r n 2 = r n 1 q n + r n. If r n = 0, we say that b is an n-stage divisor of a or b n-divides a, written b n a. The ring R is n-quasi-euclidean if b n a for nonzero a, b R, and is quasi-euclidean if for nonzero a, b R there is an m 1, depending on a and b, with b m a. The purpose of this paper is to study n-quasi-euclidean rings and quasi- Euclidean rings and their relation to matrix reduction. In Section 2, we discuss matrix reduction and some rings related to it. In Section 3, we study n-quasi- Euclidean rings and quasi-euclidean rings. Theorem 3.3 gives a number of characterizations of quasi-euclidean rings. Now, R is 2-quasi-Euclidean, precisely when it is B ezout with stable rank 1, or, equivalently, for a, b R there is an x R with (a, b) = (a + bx). We show that R is 3-quasi-Euclidean if for REV. ROUMAINE MATH. PURES APPL. 58 (2013), 4,

2 418 D.D. Anderson and J.R. Juett 2 a, b R there exist c, d R with (a, b) = (ca + (1 cd)b) and give a similar characterization of n-quasi-euclidean rings (see Theorems 3.7 and 3.9). We end by characterizing when the monoid ring R[X; S] or power series ring R[[X]] is quasi-euclidean (Theorems 3.16 and 3.19, respectively). 2. PRELIMINARIES Throughout, all rings will be commutative with 1 0. For a ring R, we will use R to denote its group of units and abbreviate R = R \ {0}. For a ring R and n 1, the general linear group of degree n is the group GL n (R) of invertible n n matrices over R. The elementary row operations on a matrix over R are the operations of the following types: (1) multiplying a row by a unit, (2) adding a multiple of one row to another, and (3) transposing two rows. The elementary column operations are dened analogously. Instead of applying elementary row (resp., column) operations, we can equivalently left-multiply (resp., right-multiply) by appropriate invertible matrices called elementary matrices. These are the matrices of the following types: (1) invertible diagonal matrices, (2) matrices called transvections that dier from the identity matrix by one nonzero element o the main diagonal, and (3) permutation matrices, i.e., matrices obtained by permuting the rows (equivalently, columns) of the identity matrix. Matrices of type (2) correspond to operations of type (2), and matrices of types (1) and (3) correspond to compositions of operations of types (1) and (3), respectively. Following [10], we call R a GE n - ring (the GE stands for generalized Euclidean) if GL n (R) is generated by the elementary matrices. A ring is an elementary divisor ring if every matrix over it admits a diagonal reduction, i.e., for every matrix A over it there are invertible matrices P and Q with P AQ = diag(a 1,..., a n ) and each a i a i+1. (A matrix (a ij ) is diagonal if a ij = 0 for i j; we do not require a diagonal matrix to be square.) A ring is Hermite if every matrix over it admits a lower trapezoidal reduction, i.e., for every matrix A there is an invertible matrix U with AU lower trapezoidal. (A matrix (a ij ) is lower (resp., upper) trapezoidal if a ij = 0 for i < j (resp., i > j). Hence, a matrix is diagonal if and only if it is both upper and lower trapezoidal. Some sources use triangular in place of trapezoidal, but we have avoided that terminology because a triangular matrix is usually required to be square.) By symmetry, we can equivalently change AU to UA while changing the lower to upper. If R is a Hermite ring, then we can further normalize these trapezoidal reductions so that the Hermite form is achieved. Hermite rings can alternatively be characterized as the rings over which all 1 2 (resp., 2 1) matrices admit a diagonal reduction ([19], Theorem 3.5), or as

3 3 Division chains and quasi-euclidean rings 419 rings R satisfying the property that for every a, b R there are d, a, b R with a = da, b = db, and (a, b ) = R ([15], Theorem 3). For a ring to be an elementary divisor ring, it is sucient for all 1 2 (resp., 2 1) and 2 2 matrices over it to admit a diagonal reduction ([19], Theorem 5.1), and, for an integral domain, one only need consider the 2 2 matrices. An elementary reduction ring is an elementary divisor ring where the reduction can be achieved by elementary row and column operations. Equivalently, an elementary reduction ring is a GE 2 elementary divisor ring, or, equivalently, an elementary divisor ring that is GE n for all n 1. The analogous strengthening of the Hermite denition, i.e., being able to achieve the lower (resp., upper) trapezoidal reduction with elementary column (resp., row) operations, characterizes the quasi-euclidean rings that we will study in the next section. The notion of stable rank has proven useful in the study of Hermite and elementary divisor rings. The stable rank of a ring R is the inmum sr(r) of the positive integers n such that (a 1,..., a n+1 ) = R implies that there are b 1,..., b n R with (a 1 + b 1 a n+1,..., a n + b n a n+1 ) = R. It can be shown that this condition holds for n + 1 if it holds for n. We dene the almost stable rank of R to be the supremum asr(r) of the stable ranks of its proper homomorphic images, or, equivalently ([3], Proposition 4), as the inmum of the positive integers n such that a 1 0 and (a 1,..., a n+2 ) = R implies that there are b 2,..., b n+1 R with (a 1, a 2 + b 2 a n+2,..., a n+1 + b n+1 a n+2 ) = R. We note that asr(r) sr(r) asr(r) + 1. Hermite rings can be characterized as B ezout rings with stable rank at most 2 ([27], Theorem 1). (Recall that a ring is B ezout if every nitely generated ideal is principal.) Also, by ([22], Theorem 3.7), a B ezout ring with almost stable rank 1 is an elementary divisor ring, but [24] gives Z + XQ[X] as an example of an elementary divisor domain (in fact, an elementary reduction ring) with almost stable rank 2. It is easily shown that if I is an ideal contained in the Jacobson radical J (R) of R, then sr(r) = sr(r/i). (However, the inequality asr(r) asr(r/i) can be strict. For example, we have asr(z[[x]]) = 2 and asr(z[[x]]/j (Z[[X]])) = asr(z) = 1.) From this observation, we immediately deduce two noteworthy consequences: (A) If R is a B ezout ring and I is an ideal contained in its Jacobson radical, then R is Hermite if and only if R/I is. (B) A B ezout ring whose Jacobson radical contains a prime ideal is Hermite. A ring R is von Neumann regular if it satises one of the following equivalent conditions: (1) for every a R there is an x R with a 2 x = a, (2) every element is a unit times an idempotent, (3) every principal ideal is generated by an idempotent, (4) every nitely generated ideal is a principal ideal

4 420 D.D. Anderson and J.R. Juett 4 generated by an idempotent, or (5) R is zero-dimensional and reduced. In addition to semihereditary rings being arithmetical, we have the following implications: von Neumann regular B ezout with stable rank 1 elementary reduction ring elementary divisor ring Hermite B ezout arithmetical Gaussian Pr ufer, and none of the implications reverse. (Recall that a ring is arithmetical if its lattice of ideals is distributive, and is Pr ufer if every nitely generated regular ideal is invertible. Finally, the ring R is Gaussian if for f, g R[X] we have c(fg) = c(f)c(g), where c(f) denotes the content of f.) In the case of pr esimpliable rings (rings with all zero divisors contained in the Jacobson radical, or, equivalently, rings where xy = x x = 0 or y is a unit), the Hermite and B ezout properties are equivalent ([19], Theorem 3.2) (this is generalized by observation (B) above), and the arithmetical, Gaussian, and Pr ufer properties are equivalent ([20], Theorem 64) (slightly augment the proof to obtain the pr esimpliable generalization). In the case of domains, we can add semihereditary to the second equivalence. It remains an open question whether there is a B ezout domain that is not an elementary divisor domain. We refer the reader to [17] for information on the aforementioned rings between Pr ufer and semihereditary rings, and to [19] for information on B ezout, Hermite, and elementary divisor rings. 3. DIVISION CHAINS Let R be a ring. Following Cooke [11], for a, b R and n 1, we dene an n-stage division chain starting with the pair (a, b) and ending with r n to be a sequence of equations (*): a = bq 1 + r 1, b = r 1 q 2 + r 2, r 1 = r 2 q 3 + r 3,... r n 2 = r n 1 q n + r n. In this case, we will write DC(a, b, r n, n) or DC(a, b, r n ). For convenience, we will sometimes take r 1 = a and r 0 = b. A division chain is terminating if it ends with 0, and minimal if there is no shorter division chain with the same start and end. We say b is an n-stage divisor of a or b n-divides a, written b n a, if DC(a, b, 0, n), i.e., if there is a terminating n-stage division chain starting with (a, b). We dene length(a, b, r) = inf{n DC(a, b, r, n)} and length(a, b) = length(a, b, 0). (Note that length(a, b, r) = if DC(a, b, r) does not hold.) We call b a strict n-stage divisor of a if length(a, b) = n <, or, equivalently, there is a minimal terminating n-stage division chain starting

5 5 Division chains and quasi-euclidean rings 421 with (a, b). We dene the diameter of R to be diam(r) = sup{length(a, b) a, b R } = sup{length(a, b) a R, b R }. We next collect some simple facts about division chains and the n relation. Theorem 3.1. Let R be a ring, a, b, d, r R, and n 1. (1) DC(a, b, r, 1) a r (b). (2) If DC(a, b, r, n) as in (*), then DC(r i 1, r i, r, n i) for 0 i n 1. (3) DC(a, b, r, n) DC(a, b, r, n+1) and DC(b, a, r, n+1). Hence, length(b, a, r) length(a, b, r) + 1. (4) DC(a, b, r, n) DC(ad, bd, rd, n). (5) If d is a regular common divisor of a and b, then: (a) DC(a, b, r, n) DC( a d, b d, r d {, n). length( a (b) length(a, b, r) = d, b d, r d ), d r, d r. (6) Let u, v R. Then DC(a, b, r, n) DC(ua, vb, ur, n) if n is odd and DC(ua, vb, vr, n) if n is even. Hence, DC(a, b, r, n) DC(ua, vb, ur, n + 1) and DC(ua, vb, vr, n + 1). (7) For a division chain as in (*), we have (a, b) = (b, r 1 ) = (r 2, r 3 ) = = (r n 1, r n ). (8) { 1 + inf{length(b, x, r) a x (b)}, a r / (b) length(a, b, r) = 1, a r (b). Proof. Parts (1), (2), and (7) are clear, and part (8) follows from parts (1) and (2). We now prove the remaining parts. (3) Assume DC(a, b, r, n) as in (*). Replacing the last equation in (*) with the two equations r n 2 = r n 1 (q n + 1) + (r n r n 1 ) and r n 1 = (r n r n 1 )( 1)+r n, we see that DC(a, b, r, n+1). Inserting b = a 0+b before (*), we see that DC(b, a, r, n + 1). (4) Multiply the equations in (*) by d. (5) For part (a), divide the equations in (*) by d, noting that d divides each r i. For part (b), the d r case follows from (4) and (5a), while the d r case follows from (7) and the fact that (a, b) (d). (6) In view of (3), it will suce to show the rst statement. Given DC(a, b, r, n) as in (*), we reach our desired conclusion by writing ua = (vb)(uv 1 q 1 ) + ur 1, vb = (ur 1 )(vu 1 q 2 )+vr 2, ur 1 = (vr 2 )(uv 1 q 3 )+ur 3, and continuing in this fashion.

6 422 D.D. Anderson and J.R. Juett 6 Corollary 3.2. Let R be a ring, a, b, d, r R, and n 1. (1) a 1 b a b. (2) If b n a as in (*), then r i n (i+1) r i 1 for 1 i n 1. (3) b n a a n+1 b and b n+1 a. Hence, length(b, a) length(a, b) + 1. (4) (a) b n 0. (b) 0 n a a = 0 or n 2. Hence, length(0, b) = 1 and length(a, 0) = 2 for a 0. (5) b n a bd n ad. (6) If b n a and d is a regular common divisor of a and b, then b d n a d. Hence, if d is a regular common divisor of a and b, then length(a, b) = length( a d, b d ). (7) For u, v R, a n b ua n vb, and length(a, b) = length(ua, vb). (8) If b n a as in (*), then (a, b) = (r n 1 ). Moreover, in this case, if (a, b) = (d), then there are a, b R with a = da, b = db, and (a, b ) = R. (9) b 2 a if and only if there is an x R with (a + bx) = (a, b). Hence, if (a, b) = R, then b 2 a if and only if there is an x R with a + bx a unit. (10) { 1 + inf{length(b, r) a r (b)}, b a length(a, b) = 1, b a. Proof. We prove parts (8) and (9). The rest are immediate from Theorem 3.1. (8) The rst statement is ([11], Proposition 3), but we can also immediately get it from Theorem 3.1 part (7). For the moreover statement, by ([15], Lemma 4) it will suce to consider the case d = r n 1. If n = 1, then a = r n 1 q 1, b = r n 1 1, and (q 1, 1) = R. So let us assume n 2. Then (*) gives r 1 n 1 b, so by induction we have b = r n 1 a and r 1 = r n 1 b for some a, b R with (a, b ) = R. Then a = bq 1 + r 1 = r n 1 (a q 1 + b ), b = r n 1 a, and (a q 1 + b, a ) = (a, b ) = R. (9) ( ): If b 2 a as in (*), then (a, b) = (r 1 ) = (a bq 1 ). ( ): If there is an x R with (a + bx) = (a, b), then a = b( x) + (a + bx) and a + bx b, showing that b 2 a. For n 1 and a function φ from R into a well-ordered set W, we call R an n-stage (W -)Euclidean ring with respect to φ if for every a, b R there is an n-stage division chain as in (*) with φ(r n ) < φ(b). A (W -)Euclidean ring is a 1-stage (W -)Euclidean ring. (Cooke [11] restricts the denitions to the case W = ω = {0, 1, 2,...}, so his (n-stage) Euclidean ring corresponds to

7 7 Division chains and quasi-euclidean rings 423 what we would call an (n-stage) ω-euclidean ring. We have expanded the denition so that a 1-stage Euclidean ring corresponds to a Euclidean ring in the sense of Samuel [25].) By Corollary 3.2, one can equivalently replace the n-stage division chains with division chains of at most n stages in the denition of n-stage (W -)Euclidean. Hence, n-stage (W -)Euclidean (n + 1)-stage (W -)Euclidean. Let R be a ring. For n 1, we call R (strictly) n-quasi-euclidean if for each a, b R we have b n a (and there are a 0, b 0 R with b 0 strictly n- dividing a 0 ). For n 2, we can equivalently replace the R with R in these denitions. Equivalently, an n-quasi-euclidean ring is a ring with diameter at most n, and a strictly n-quasi-euclidean ring is a ring with diameter n. We note that diam(r) = inf{n R is n-quasi-euclidean}. Of course, an n-quasi- Euclidean ring is n-stage ω-euclidean with respect to the function φ : R ω given by φ(0) = 0 and φ(a) = 1 for a 0. Following Chen [9], we call a ring R stably Euclidean if length(a, b) < for every comaximal a, b R (equivalently, R ), or, equivalently ([9], Theorem ), if every 1 2 (resp., 2 1) matrix over R with comaximal entries admits a diagonal reduction by elementary column (resp., row) operations, or, equivalently ([9], Theorem ), if R is GE 2 and every 1 2 (resp., 2 1) matrix over R with comaximal entries can be completed to an invertible 2 2 matrix. Theorem 3.3. The following are equivalent for a ring R. (1) length(a, b) < for every a, b R (resp., a, b R) (2) There is a well-ordered set W and a function φ : R W such that for every a, b R we have DC(a, b, r) for some r R with φ(r) < φ(b). (3) There is a function φ : R ω such that for every a, b R we have DC(a, b, r) for some r R with φ(r) < φ(b). (4) There is a well-ordered set W and a function φ : R R W such that for every a, b R there are q, r R with a = bq +r and φ(b, r) < φ(a, b). (5) There is a function φ : R R ω such that for every a R and b R there are q, r R with a = bq + r and φ(b, r) < φ(a, b). (6) Every 1 2 (resp., 2 1) matrix over R admits a diagonal reduction by elementary column (resp., row) operations. (7) Every matrix over R admits a lower (resp., upper) trapezoidal reduction by elementary column (resp., row) operations. (8) The ring R is Hermite and GE 2. (9) The ring R is Hermite and GE n for all n 1. (10) The ring R is Hermite and stably Euclidean.

8 424 D.D. Anderson and J.R. Juett 8 Proof. Note rst that the dierent forms of (6) and (7) are equivalent by symmetry, and the two forms of (1) are equivalent since length(a, b) is 1 or 2 when a or b is zero. We have (1) (3) by ([11], Proposition 1), and a small adjustment of the proof gives (1) (2). The equivalence (3) (4) (6) is from ([4], Theorem 5, Proposition 23). In ([11], Proposition 4) it is shown that (3) implies R is GE n for all n 1, and, since (6) clearly implies that R is Hermite, we get (3) (9). The equivalence (3) (10) is ([9], Corollary ). Because (9) (8) (6) (7) and (5) (4) are clear, and (10) (7) can be shown by trivial adjustments to the proof of ([9], Corollary ), all that remains is to show (1) (5). Assume length(a, b) < for every a, b R. Dene φ : R R ω by φ(a, 0) = 0 and φ(a, b) = length(a, b) for b 0. Take any a R and b R. If b a, then by Corollary 3.2 part (10) we have φ(a, b) = 1 + inf{length(b, r) a r (b)} = 1 + min{φ(b, r) a r (b)}, so we may write a = bq + r for some q, r R with φ(b, r) < φ(a, b). On the other hand, if b a, then a = b( a b ) + 0 and φ(a, b) = length(a, b) = 1 > 0 = φ(b, 0). Following Bougaut [4], we call a ring R satisfying the equivalent conditions of Theorem 3.3 quasi-euclidean. (There are other names for such a ring. O'Meara [23] says that such a ring satises the Euclidean chain condition, and Cooke [11] calls such a ring ω-stage Euclidean.) It is immediate from (6) or (7) that elementary reduction ring quasi-euclidean Hermite, and from (2) we see that n-quasi-euclidean n-stage Euclidean quasi-euclidean. However, Bougaut [4] gives R[X, Y ]/(X 2 + Y 2 + 1) as an example of a PID (hence, elementary divisor domain) that is not quasi-euclidean (in fact, is not stably Euclidean). Another example, noted by Cooke [11], is the ring of algebraic integers in Q( 19). An example of a stably Euclidean ring that is not quasi- Euclidean is a quasilocal ring that is not B ezout. The paper [8] constructs examples of quasi-euclidean domain that are not 2-stage Euclidean, such as Z + XQ[X] and K[X] + Y K(X)[Y ], where K is a nite eld. It is well known that a(n) (ω-)euclidean domain is a PID, and ([14], Theorem 5.3) shows that the converse is true if the stable rank is 1. However, there are examples of ω-euclidean domains with stable rank 2, such as Z or K[X], where K is a eld. Theorem 3.4. Let R be a ring and φ : R R ω. The following are equivalent. (1) The ring R is quasi-euclidean and φ is the unique smallest function satisfying Theorem 3.3(5). (2) (a) φ(a, b) = 0 b = 0, and (b) for a R and b R we have φ(a, b) = 1+min{φ(b, r) a r (b)}.

9 9 Division chains and quasi-euclidean rings 425 (3) The ring R is quasi-euclidean, φ(a, 0) = 0, and φ(a, b) = length(a, b) for b 0. In this case, we have diam(r) = φ(r R ). Proof. (3) (2): See the proof of (1) (5) in Theorem 3.3. (2) (1): Assume (2) and let ψ : R R ω be any function as in Theorem 3.3(5). Take any a, b R. If ψ(a, b) = 0, then Theorem 3.3(5) implies that b = 0, and hence, φ(a, b) = 0 = ψ(a, b). So let us assume ψ(a, b) 1. If b = 0, then φ(a, b) = 0 ψ(a, b), so let us assume b 0. Then there are q, r R with a = bq + r and ψ(b, r) < ψ(a, b). By induction, we have φ(b, r) ψ(b, r), so ψ(a, b) ψ(b, r) + 1 φ(b, r) + 1 φ(a, b). (1) (3): Follows from (3) (1) and uniqueness. The last statement is clear from (3) and the observation that (2) implies that φ(r R) is an interval. Theorem 3.5. (1) An n-quasi-euclidean ring is m-quasi-euclidean for m n. (2) A ring is 1-quasi-Euclidean if and only if it is a eld. (3) The following are equivalent for a ring R. (a) R is 2-quasi-Euclidean. (b) R is B ezout with stable rank 1. (c) For every a, b R, there is an x R with (a, b) = (a + bx). (4) The following are equivalent for a ring R and a nonempty family of algebraically independent indeterminates {X λ } λ Λ. (a) R is arithmetical. (b) R({X λ }) is arithmetical. (c) R({X λ }) is B ezout with stable rank 1. (5) A principal ideal ring with stable rank 1 is Euclidean. Proof. (1) Follows from Corollary 3.2 part (3). (2) By Corollary 3.2 part (1), a ring R is 1-quasi-Euclidean if and only if a b for every a, b R. The result is now clear. (3) (We note that (b) (c) for domains is ([14], Proposition 5.1).) (a) (c): Corollary 3.2 part (9). (c) (b): Clear.

10 426 D.D. Anderson and J.R. Juett 10 (b) (a): Assume R is B ezout with stable rank 1. Take any a, b R. Since R is Hermite, we may write a = a d and b = b d, where (a, b ) = R. Since sr(r) = 1, there is an x R with (a + b x) = R. So, by Corollary 3.2 part (9), we obtain b 2 a, and from Corollary 3.2 part (5) we deduce b 2 a. (4) We have (a) (b) R({X λ }) is B ezout by ([1], Theorem 8(1)), and (c) (b) is clear. So it will suce to show that R({X λ }) has stable rank 1. Assume ( f h, g h ) = R({X λ}), where f, g, h R[{X λ }] with c(h) = R. Then R({X λ }) = (f, g), so R({X λ }) = c(f)r({x λ }) + c(g)r({x λ }) = c(f + Xα N g)r({x λ }) for N large enough, where α Λ. Therefore c(f +Xα N g) = R, and hence f + Xα N g is a unit in R({X λ }). Therefore f h + ( g h )XN α = 1 h (f + XN α g) is a unit, as desired. (5) Recall that a PIR is a nite direct product of PID's and SPIR's (where a special principal ideal ring or SPIR is a local Artinian PIR). Of course, a SPIR is Euclidean, and a PID with stable rank 1 is Euclidean by ([14], Theorem 5.3), so the result follows from the fact that a nite direct product of Euclidean rings is Euclidean. Theorem 3.5 part (3) gives us several examples of 2-quasi-Euclidean rings, such as semi-quasilocal B ezout rings, zero-dimensional B ezout rings, or the rings R({X λ }) where R is arithmetical. However, recall that a PID (hence, a one-dimensional B ezout ring) need not be quasi-euclidean. The following construction gives a 2-quasi-Euclidean ring with an arbitrary lattice-ordered group as its group of divisibility. Example 3.6. (Every lattice-ordered abelian group is the group of divisibility of a 2-quasi-Euclidean domain, i.e., a B ezout domain with stable rank 1.) Let (G, ) be a lattice-ordered abelian group, K be a eld, and K[X; G] be the group ring of G over K. Dene w : K[X; G] G { } by w(0) = and w( n i=1 a ix g i ) = inf{g i } n i=1. Then w extends to a semivaluation w : L G { }, where L is the quotient eld of K[X; G]. Let D = w 1 (G + { }). Heinzer [18] has shown that D is B ezout with stable rank 1. We note that having stable rank 1 is not an ideal-theoretical property. Let R be a B ezout domain with quotient eld K. So its group of divisibility G = K /R is a lattice-ordered group. Let D be the B ezout domain constructed in Example 3.6; so D has stable rank 1. Let L(R) and L(D) be the lattice of ideals of R and D, respectively. Then the map θ : L(R) L(D) given by θ(0) = 0 and θ(i) = (X ir i I \{0}) is a multiplicative lattice isomorphism. Here, D has stable rank 1 while R need not.

11 11 Division chains and quasi-euclidean rings 427 The characterizations of 2-quasi-Euclidean rings given in Theorem 3.5 part (3) have an analog for n-quasi-euclidean rings for n 3. We next explicitly handle the case n = 3 and then formulate the general case. Theorem 3.7. The following are equivalent for a ring R. (1) R is 3-quasi-Euclidean. (2) For a, b R, there exist c, d R with (a, b) = (ca + (1 cd)b). (3) R is Hermite and (2) holds for comaximal a, b R. Proof. (1) (3): Let R be 3-quasi-Euclidean and a, b R. Then R is Hermite, and we can write a = bq + r and b = rq + r, where r r. Then r = b rq = b (a bq)q = q a+(1+q q)b, so (a, b) = (r ) = (ca+(1 cd)b), where c = q and d = q. (3) (2): Assume (3) and let a, b R. Since R is Hermite, there are a, b, x R with a = a x, b = b x, and (a, b ) = R. By (3), there are c, d R with (a, b ) = (ca + (1 cd)b ), and hence (a, b) = (ca + (1 cd)b). (2) (1): Assume (2) and let a, b R. Then there are c, d R with (a, b) = (ca + (1 cd)b). Let r 2 = ca + (1 cd)b and r 1 = a bd. Then a = bd + r 1, b = bcd ca + r 2 = r 1 ( c) + r 2, and r 2 r 1, so b 3 a. We now extend Theorem 3.7 to the general case. We recursively dene a sequence of polynomials {α n } n=0 Z[X 1, X 2,...] by α 0 = 0, α 1 = 1, and α n = α n 2 X n 1 α n 1 for n 2. It is easily seen that α n Z[X 1,..., X n 1 ] and deg α n = n 1 for n 1. Lemma 3.8. Let R be a ring, a, b R, and n 1. Then b n a if and only if we can write (a, b) = (α n 1 (c 1,..., c n 2 )a + α n (c 1,..., c n 1 )b). Proof. We have already established the cases n = 1 and n = 2, so we may assume n 3. ( ): Assume b n a. Then there are q, r R with a = bq + r and r n 1 b. By induction, there are c 1,..., c n 2 R with (b, r) = (α n 2 (c 1,..., c n 3 )b + α n 1 (c 1,..., c n 2 )r), and thus: (a, b) = (b, r) = (α n 1 (c 1,..., c n 2 )a + (α n 2 qα n 1 )(c 1,..., c n 2 )b) = (α n 1 (c 1,..., c n 2 )a + α n (c 1,..., c n 2, q)b). ( ): Assume we can write (a,b)=(α n 1 (c 1,..., c n 2 )a+α n (c 1,..., c n 1 )b). Then: (b, a bc n 1 ) = (a, b) = (α n 1 (c 1,..., c n 2 )a + (α n 2 c n 1 α n 1 )(c 1,..., c n 2 )b) = (α n 2 (c 1,..., c n 3 )b + α n 1 (c 1,..., c n 2 )(a bc n 1 )),

12 428 D.D. Anderson and J.R. Juett 12 so, by induction we have a bc n 1 n 1 b. But a = bc n 1 + (a bc n 1 ), so b n a. Theorem 3.9. The following are equivalent for a ring R and n 2. (1) R is n-quasi-euclidean. (2) For a, b R, there are c 1,..., c n 1 R with (a, b) = (α n 1 (c 1,..., c n 2 )a + α n (c 1,..., c n 1 )b). (3) R is Hermite and (2) holds for comaximal a, b R. Proof. We have (1) (2) by Lemma 3.8, and it is clear that (1) and (2) together imply (3), so it will suce to show (3) (2). Assume (3) and let a, b R. Since R is Hermite, there are a, b, d R with a = a d, b = b d, and (a, b ) = R. By (3), there are c 1,..., c n 1 R with (a, b ) = (α n 1 (c 1,..., c n 2 )a + α n (c 1,..., c n 1 )b ), and multiplying by d nishes the proof. Proposition Homomorphic images, localizations, and overrings of (n-)quasi-euclidean rings are (n-)quasi-euclidean. Furthermore, when passing from a ring to a homomorphic image or localization, the diameter is not increased. Proof. The last statement will follow from the proof of the rst. Overrings of B ezout rings are localizations, so we only need to show the homomorphic image and localization parts. The former is easily shown by applying a given homomorphism to the equations in (*). We will demonstrate the latter by showing that, given a ring R, a multiplicatively closed subset S of R, s, t S, and a, b R with a m b in R, we have a s m b t in R S. In this case, we certainly have a 1 m b 1 in R S, and we can then apply Corollary 3.2 part (7) to reach the desired conclusion. Proposition Let {R λ } λ Λ be a nonempty family of rings. (1) If Λ 2, then diam( R λ ) = sup λ max(diam(r λ ), 2). Hence, if n 2 and each R λ is n-quasi-euclidean, then R λ is n-quasi-euclidean. (2) If R λ is n-quasi-euclidean (resp., quasi-euclidean), then so is each R λ. Moreover, if R λ is strictly n-quasi-euclidean, then at least one R λ is strictly n-quasi-euclidean. (3) If Λ is nite, then R λ is quasi-euclidean if and only if each R λ is. Proof. Follows from the denitions applied to each coordinate, or from a suitable use of Theorem 3.4. We leave the details to the reader. Lemma Let R be a ring and I be an ideal contained in its Jacobson radical. The following are equivalent for comaximal a, b R.

13 13 Division chains and quasi-euclidean rings 429 (1) b n a in R. (2) b + I n a + I in R/I. (3) b + x n a + y for every x, y I. Proof. We abbreviate R = R/I and x = x + I for x R. The implication (3) (1) (2) is clear. Now assume b n ā. If n = 1, then R b = Rā+ R b = R, so Rb = R and hence, b + x is a unit for every x I. So let us assume n 2. Then we may write ā = b q + r for some q, r R with r n 1 b. Write a = bq + r + z for some z I. Then, for every x, y I we have a+y = (b+x)q +(r +(y +z qx)), and r +(y +z qx) n 1 b+x by induction, and thus b + x n a + y. McGovern ([22], Corollary 2.3) shows that a B ezout ring R is an elementary divisor ring if and only if R/J (R) is. (Examining the proof shows that J (R) can actually be replaced with any ideal I J (R).) We have analogous results for stably Euclidean rings, quasi-euclidean rings, and elementary reduction rings. Corollary Let R be a ring and I be an ideal contained in its Jacobson radical. Then R is stably Euclidean if and only if R/I is. Theorem Let R be a B ezout ring and I be an ideal contained in its Jacobson radical. (1) The ring R is quasi-euclidean if and only if R/I is. (2) diam(r/i) diam(r) max(2, diam(r/i)). Proof. (1) This is immediate from Corollary 3.13 and the fact that R is Hermite if and only if R/I is, but we will also give an alternate proof in order to facilitate the proof of part (2). ( ): Proposition ( ): Assume R/I is quasi-euclidean. Take any a, b R. Since R/I is Hermite, the same holds for R, so there are a, b, d R with a = da, b = db, and (a, b ) = R. We get length(a, b) length(a, b ) = length(a + I, b + I) < by Corollary 3.2 part (5) and Lemma (2) We get diam(r) diam(r/i) by Proposition 3.10, and from the proof of part (1) we obtain diam(r) max(2, diam(r/i)). Corollary Let R be a B ezout ring and I be an ideal contained in its Jacobson radical. Then R is an elementary reduction ring if and only if R/I is.

14 430 D.D. Anderson and J.R. Juett 14 Proof. ( ): Homomorphic images of elementary reduction rings are elementary reduction rings. ( ): Since elementary reduction rings are precisely the quasi-euclidean elementary divisor rings, this follows from Theorem 3.14 and the aforementioned result that R is an elementary divisor domain if and only if R/I is. By a monoid, we mean an additive commutative semigroup with identity. We denote the group of units of a monoid S by S, and we abbreviate S = S\{0}. Given a ring R and a monoid S, we construct the monoid ring R[X; S] = {a 1 X s a n X sn a 1,..., a n R, s 1,..., s n S} with addition and multiplication dened analogously to polynomial addition and multiplication. For example, we have R[X; Z + ] = R[X] and R[X; Z] = R[X, X 1 ]. A monoid S is torsion-free if ns = nt s = t for n 1 and s, t S, and cancellative if a + b = a + c b = c for a, b, c S. A monoid is Pr ufer if it is an ascending union of cyclic submonoids, where a monoid is cyclic if it is generated (as a monoid) by a single element. An excellent resource for monoid rings is [16]. The article [5] has a discussion on the history of the following theorem in the special case of polynomial rings, i.e., when S = Z +. Theorem The following are equivalent for a ring R and a nonzero torsion-free cancellative monoid S. (1) R is von Neumann regular and S is (up to isomorphism) a subgroup of Q or a Pr ufer submonoid of Q +. (2) Every 2-generated regular ideal of R[X; S] is invertible. (3) R[X; S] is Pr ufer. (4) R[X; S] is Gaussian. (5) R[X; S] is arithmetical. (6) R[X; S] is semihereditary. (7) R[X; S] is B ezout. (8) R[X; S] is Hermite. (9) R[X; S] is quasi-euclidean. (10) R[X; S] is an elementary divisor ring. (11) R[X; S] is an elementary reduction ring. (12) R[X; S] is one-dimensional and reduced, and S is a group or Pr ufer monoid. Proof. The equivalence (1) (3) (5) (7) is given in ([16], Theorem 18.9), and of course (6) (5) and (11) (7) (5) (2) is true with any ring in place of R[X; S], so it will suce to show (2) (1) (11) and (6) (1) (12). (The original version of ([16], Theorem 18.9) states that

15 15 Division chains and quasi-euclidean rings 431 S is a subgroup or Pr ufer submonoid of Q in (1), but this is equivalent to the way we have stated it since ([16], Theorem 2.9) asserts that a submonoid of Q containing both positive and negative elements is a group.) (2) (1): In ([16], Theorem 18.9), it is shown that (1) holds if (f, g) 2 = (f 2, g 2 ) for every f, g R[X; S] with f or g regular. This condition holds if every 2-generated regular ideal of R[X; S] is invertible. (See ([5], Remark 10).) (1) (11): Assume (1). Two dierent proofs that R[X] is an elementary divisor ring are given in ([26], Theorem) and ([6], Theorem 2). Slightly adapting the proof of the latter shows that R[X] is in fact an elementary reduction ring. (An alternate way to do this would be to combine ([6], Theorem 2) or ([26], Theorem) with ([12], Theorem 1.3), where the latter asserts that a polynomial ring is GE 2 if and only if the base ring is zero-dimensional. Since the proof of the latter result is relatively dicult, a more elementary method would be to replace ([12], Theorem 1.3) by Proposition 3.18 below, which shows that a polynomial ring over a von Neumann regular ring is quasi-euclidean.) The result now follows after we make three simple observations: (i) the property of being an elementary reduction ring is preserved by localization, (ii) an ascending union of elementary reduction rings is an elementary reduction ring, and (iii) the hypotheses on S ensure that R[X; S] is an ascending union of rings that are isomorphic to R[X] or R[X, X 1 ]. (1) (6): Assume (1). We recall that Endo's Theorem [13] shows that semihereditary rings are precisely those arithmetical rings whose total quotient rings are von Neumann regular, so, in view of (1) (5), all that remains is to show that T (R[X; S]) is von Neumann regular. The hypotheses on S ensure that we may write it as an ascending union of submonoids S = λ Λ S λ, where each S λ is isomorphic to Z or Z +. By ([16], Corollary 8.6), the zero divisors of each R[X; S λ ] are precisely those elements whose coecients are all annihilated by a single element. So an element of R[X; S λ ] is regular if and only if it is regular as an element of R[X; S]. It now follows that T (R[X; S]) = λ Λ T (R[X; S λ]), where each of the latter total quotient rings is isomorphic to T (R[X]) = T (R[X, X 1 ]). Because McCarthy [21] has shown that R[X] is semihereditary, each of these is von Neumann regular, and thus T (R[X; S]) is von Neumann regular. (1) (12): From ([16], Theorems 17.1 and 21.4] we obtain dim R[X; S] = dim R[X; G] = dim R[X 1,..., X n ], where G is the quotient group of S and n is the torsion-free rank of G, i.e., the dimension of G Z as a vector space over Q. Note that a torsion-free group has torsion-free rank 1 if and only if it is (isomorphic to) a subgroup of Q. Therefore R[X; S] is one-dimensional R is zero-dimensional and n = 1 R is zero-dimensional and S is (up to isomorphism) a submonoid of Q. By ([16], Theorem 9.17), we see that

16 432 D.D. Anderson and J.R. Juett 16 R[X; S] is reduced if and only if R is. Since von Neumann regular rings are precisely the zero-dimensional reduced rings, combining the last two sentences gives (1) (12). The proof of (1) (12) in Theorem 3.16 shows that, if R is a ring and S is a torsion-free cancellative monoid, then R[X; S] is one-dimensional (and reduced) if and only if R is zero-dimensional (von Neumann regular) and S is isomorphic to a submonoid of Q. We state Theorem 3.16 again for the special case of domains, where we can add one more equivalent statement. Note that the hypotheses on D and S at the start of the following corollary are precisely what is required for D[X; S] to be an integral domain ([16], Theorem 8.1). Corollary The following are equivalent for an integral domain D and a nonzero torsion-free cancellative monoid S. (1) D is a eld and S is (up to isomorphism) a subgroup of Q or a Pr ufer submonoid of Q +. (2) D[X; S] is Pr ufer. (3) D[X; S] is B ezout. (4) D[X; S] is quasi-euclidean. (5) D[X; S] is an elementary divisor domain. (6) D[X; S] is an elementary reduction domain. (7) D[X; S] is one-dimensional and S is a group or Pr ufer monoid. (8) D[X; S] is B ezout with almost stable rank 1. Proof. Since (8) (2) is clear, and (1)(7) are equivalent by Theorem 3.16, we only need to show (1) (8). Assume (1). The hypotheses on S ensure that D is an ascending union of rings that are isomorphic to D[X] or D[X, X 1 ]. Each of these is a PID (in fact an ω-euclidean domain), and thus, has almost stable rank 1. (In [3] it is shown that, for n 1, an n-dimensional Noetherian domain has almost stable rank at most n.) With the observation that an ascending union of rings each having almost stable rank 1 has almost stable rank 1, we nish the proof. Note that the analog of (8) above cannot be added to Theorem 3.16 since sr(r) = asr(r) for a ring R with zero divisors by ([3], Theorem 3), but a polynomial ring never has stable rank 1. As alluded to in the proof of Theorem 3.16, there is a direct proof that a ring R is von Neumann regular if and only if R[X] is quasi-euclidean. In fact, we have the following slightly stronger result that has some interest in its own right.

17 17 Division chains and quasi-euclidean rings 433 Proposition Let R be a von Neumann regular ring, f, g R[X], and n = max(deg g, 0). Then g n+2 f. Proof. First, we consider the case where n = 0. Then g R, so there is an s R with g 2 s = g. Let r 1 = g + (1 gs)f and q 1 = (fs 1)sr 1. Then gsr 1 = g and gq 1 + r 1 = g(fs 1) + g + (1 gs)f = f, so g 2 f. Now assume n 1. Let b be the leading coecient of g, and pick s R with b 2 s = b. Since deg (1 bs)g < n, we have (1 bs)g n+1 (1 bs)f by induction. Hence, (1 bs)g n+2 (1 bs)f, say: (1 bs)f = (1 bs)gq 1 + r 1, (1 bs)g = r 1 q 2 + r 2, r 1 = r 2 q 3 + r 3,... r n = r n+1 q n+2 + r n+2, where r n+2 = 0. The leading coecient of bsg is b, which is a unit in bsr[x], so there are q, r bsr[x] with bsf = (bsg)q + r and deg r < deg bsg n. By induction, we have r n+1 bsg in R[X], so bsg n+2 bsf, say: bsf = (bsg)q 1 + r 1, bsg = r 1 q 2 + r 2, r 1 = r 2 q 3 + r 3,... r n = r n+1 q n+2 + r n+2, where r n+2 = 0. Note that 1 bs (resp., bs) divides each r i (resp., r i ), and if necessary, we can modify our construction so that 1 bs (resp., bs) divides each q i (resp., q i ). Hence, each r iq j = r i q j = 0. Dene q k = q k + q k and r k = r k + r k for k = 1,..., n + 2, where r 1 = (1 bs)f, r 0 = (1 bs)g, r 1 = bsf, and = bsg. Then for k = 1,..., n we have: r 0 r k+1 q k+2 + r k+2 = (r k+1 + r k+1 )(q k+2 + q k+2 ) + (r k+2 + r k+2 ) = (r k+1 q k+2 + r k+2 ) + (r k+1 q k+2 + r k+2 ) = r k + r k = r k. Since r 1 = (1 bs)f + bsf = f, r = 0, this shows that g n+2 f. = (1 bs)g + bsg = g, and r n+2 = Theorem The following are equivalent for a ring R. (1) R is von Neumann regular and ℵ 0 -algebraically compact. (2) R[[X]] is Gaussian. (3) R[[X]] is arithmetical.

18 434 D.D. Anderson and J.R. Juett 18 (4) R[[X]] is B ezout. (5) R[[X]] is Hermite. (6) R[[X]] is quasi-euclidean. (7) R[[X]] is an elementary divisor ring. (8) R[[X]] is an elementary reduction ring. (9) R[[X]] is B ezout with stable rank 1. Proof. Because (9) (2) holds with any ring in place of R[[X]], and since ([2], Theorem 17) gives the equivalence of (1) (4), it will suce to show (1) sr(r) = 1. But any von Neumann regular ring has stable rank 1, and sr(r[[x]]) = sr(r) since (X) J (R[[X]]) and R[[X]]/(X) = R, so the proof is complete. We note that ([7], Example 2) shows that the statement R[[X]] is semihereditary is not equivalent to the above statements. Corollary The following are equivalent for an integral domain D. (1) D is a eld. (2) D[[X]] is Pr ufer. (3) D[[X]] is B ezout. (4) D[[X]] is quasi-euclidean. (5) D[[X]] is an elementary divisor domain. (6) D[[X]] is an elementary reduction domain. (7) D[[X]] is a discrete valuation ring. REFERENCES [1] D.D. Anderson, Multiplication ideals, multiplication rings, and the ring R(X). Canad. J. Math. 28 (1976), [2] D.D. Anderson and V. Camillo, Armendariz rings and Gaussian rings. Comm. Algebra 26 (1998), [3] D.D. Anderson and J.R. Juett, Stable range and almost stable range. J. Pure Appl. Algebra 216 (2012), [4] B. Bougaut, Anneaux quasi-euclidiens. PhD Thesis, Universit e de Poitiers, [5] J.W. Brewer, A polynomial ring sampler. In A. Facchini, E. Houston and L. Salce (Eds.), Rings, Modules, Algebras, and Abelian Aroups. Marcel Dekker, New York, 2004, pp [6] J.W. Brewer, D. Katz and W. Ullery, Pole assignability in polynomial rings, power series rings, and Pr ufer domains. J. Algebra 106 (1987), [7] J.W. Brewer, E.A. Rutter and J.J. Watkins, Coherence and weak global dimension of R[[X]] when R is von Neumann regular. J. Algebra 46 (1977),

19 19 Division chains and quasi-euclidean rings 435 [8] C. Chen and M. Leu, The 2-stage Euclidean algorithm and the restricted Nagata's pairwise algorithm. J. Algebra 348 (2011), 113. [9] H. Chen, Rings related to stable range conditions, Series in Algebra 11, World Scientic Publishing, [10] P.M. Cohn, On the structure of the GL 2 of a ring. Inst. Haut. Etudes Sci. Publ. Math. 30 (1966), [11] G. Cooke, A weakening of the euclidean property for integral domains and applications to algebraic number theory. I. J. Reine Angew. Math. 282 (1976), [12] D.L. Costa, Zero-dimensionality and the GE 2 of polynomial rings. J. Pure Appl. Algebra 50 (1988), [13] S. Endo, On semi-hereditary rings. J. Math. Soc. Japan 13 (1961), 109. [14] D. Estes and J. Ohm, Stable range in commutative rings. J. Algebra 7 (1967), [15] L. Gillman and M. Henriksen, Some remarks about elementary divisor rings. Trans. Amer. Math. Soc. 82 (1956), [16] R. Gilmer, Commutative Semigroup Rings. University of Chicago Press, Chicago, [17] S. Glaz, Pr ufer conditions in rings with zero divisors. In: S.T. Chapman (Ed.), Arithmetical Properties of Commutative Rings and Monoids, pp CRC Press, [18] W.J. Heinzer, J-Noetherian integral domains with 1 in the stable range. Proc. Amer. Math. Soc. 19 (1968), [19] I. Kaplansky, Elementary divisors and modules. Trans. Amer. Math. Soc. 66 (1949), [20] I. Kaplansky, Commutative Rings. Allyn and Bacon, Boston, [21] P.J. McCarthy, The ring of polynomials over a von Neumann regular ring. Proc. Amer. Math. Soc. 39 (1973), [22] W.Wm. McGovern, B ezout rings with almost stable range 1. J. Pure Appl. Algebra, 212 (2008), [23] O.T. O'Meara, On the nite generation of linear groups over Hasse domains. J. Reine Angew. Math. 217 (1965), [24] M. Roitman, The Kaplansky condition and rings of almost stable range 1. Preprint. [25] P. Samuel, About Euclidean rings. J. Algebra 19 (1971), [26] T.S. Shores, Modules over semihereditary B ezout rings. Proc. Amer. Math. Soc. 46 (1974), [27] B.V. Zabavs'kyi, Reduction of matrices over B ezout rings of stable rank not higher than 2. Ukrainian Math. J. 55 (2003), Received 11 October 2012 University of Iowa, Department of Mathematics, Iowa City, IA 52242, U.S.A. dan-anderson@uiowa.edu University of Iowa, Department of Mathematics, Iowa City, IA 52242, U.S.A. jason-juett@uiowa.edu