arXiv:1609.02737v1 [] 9 Sep 2016

ON DELTA SETS AND THEIR REALIZABLE SUBSETS
IN KRULL MONOIDS WITH CYCLIC CLASS GROUPS
SCOTT T. CHAPMAN, FELIX GOTTI, AND ROBERTO PELAYO
Abstract. Let M be a commutative cancellative monoid. The set ∆(M ), which
consists of all positive integers which are distances between consecutive factorization
lengths of elements in M , is a widely studied object in the theory of nonunique
factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then
it is well-known that ∆(M ) ⊆ {1, . . . , n − 2}. Moreover, equality holds for this
containment when each class contains a prime divisor from M . In this note, we
consider the question of determining which subsets of {1, . . . , n−2} occur as the delta
set of an individual element from M . We first prove for x ∈ M that if n − 2 ∈ ∆(x),
then ∆(x) = {n − 2} (i.e., not all subsets of {1, . . . , n − 2} can be realized as delta
sets of individual elements). We close by proving an Archimedean-type property for
delta sets from Krull monoids with finite cyclic class group: for every natural number
m, there exist a Krull monoid M with finite cyclic class group such that M has an
element x with |∆(x)| ≥ m.

1. Introduction
The arithmetic of Krull monoids is a well-studied area in the theory of nonunique
factorizations. The interested reader can find a good summary of their known arithmetic properties in the monograph [14, Chapter 6]. We focus here on Theorem 6.7.1
of [14], where the authors show that
∆(M) = {1, . . . , n − 2}
for M a Krull monoid with cyclic class group of order n ≥ 3 where each class contains
a prime divisor of M. Here ∆(M) represents the set of all positive integers which are
distances between consecutive factorization lengths of elements in M.
We ask in this note a question related to the above equality that is seemingly unasked
in the literature: Which subsets T ⊆ {1, . . . , n − 2} are realized as the delta set of
an individual element in M (i.e., for which T does there exist an x ∈ M such that
T = ∆(x))? Based on the structure theorem for sets of lengths in Krull monoids with
finite class group (see [14, Chapter 4]), it is reasonable to assume that not all subsets
of {1, . . . , n − 2} will be realized. We verify this in Theorem 3.2 by showing for x ∈ M
that if n−2 ∈ ∆(x), then ∆(x) = {n−2}. We contrast this in Theorem 3.5 by showing
that we can construct delta sets of arbitrarily large size (i.e., for any m ∈ N there exists
Date: September 12, 2016.
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S. T. CHAPMAN, F. GOTTI, AND R. PELAYO

a Krull monoid M with finite cyclic class group such that M has an element x with
|∆(x)| ≥ m).
2. Definitions and Background
We open with some basic definitions from the theory of nonunique factorizations.
For a commutative cancellative monoid M, let A(M) represent the set of atoms (i.e.,
irreducible elements) of M, and M × its set of units. We provide here an informal
description of factorizations and associated notions. From a more formal point of
view, factorizations can be considered as elements in the factorization monoid Z(M),
which is defined as the free abelian monoid with basis A(M/M × ); details can be found
in the first chapter of [14].
To simplify our initial discussion, we suppose that M is reduced (i.e., has a unique
unit). We say that a1 . . . ak is a factorization of x ∈ M if a1 , . . . , ak ∈ A(M) and
x = a1 . . . ak in M. Two factorizations are equivalent if there is a permutation of
atoms carrying one factorization to the other. We denote by Z(x) ⊆ Z(M) the set
of all factorizations of x. If z ∈ Z(x), then let |z| denote the number of atoms in
the factorization z of x. We call |z| the length of z. Now, let x ∈ M \ M × with
factorizations
z = α1 · · · αt β1 · · · βs and z ′ = α1 · · · αt γ1 · · · γu
where for each 1 ≤ i ≤ s and 1 ≤ j ≤ u, βi 6= γj . Define
gcd(z, z ′ ) = α1 · · · αt
and
d(z, z ′ ) = max{s, u}
to be the distance between z and z ′ . The basic properties of this distance function can
be found in [14, Proposition 1.2.5].
An N-chain of factorizations from z to z ′ is a sequence z0 , . . . , zk such that each zi
is a factorization of x, z0 = z, zk = z ′ and d(zi , zi+1 ) ≤ N for all i. The catenary degree
of x, denoted c(x), is the minimal N ∈ N0 ∪ {∞} such that for any two factorizations
z, z ′ of x, there is an N-chain from z to z ′ . The catenary degree of M, denoted by
c(M), is defined by
c(M) = sup{c(x) | x ∈ M \ M × }.
A review of the known facts concerning the catenary degree can be found in [14,
Chapter 3]. An algorithm which computes the catenary degree of a finitely generated
monoid can be found in [6], and a more specific version for numerical monoids in [5].
We shift from considering particular factorizations to analyzing their lengths. For
x ∈ M \ M × , we define
L(x) = {n ∈ N | there are α1 , . . . , αn ∈ A(M) with x = α1 · · · αn }.

ON DELTA SETS AND THEIR REALIZABLE SUBSETS
IN KRULL MONOIDS WITH CYCLIC CLASS GROUPS

3

We refer to L(x) as the set of lengths of x in M. Further, set
L(M) = {L(x) | x ∈ M \ M × },
which we refer to as the system of sets of lengths of M. The interested reader can
find many recent advances concerning sets of lengths in [11], [18], and [19]. Given
x ∈ M \ M × , write its length set in the form
L(x) = {n1 , . . . , nk },
where ni < ni+1 for 1 ≤ i ≤ k − 1. The delta set of x is defined by
∆(x) = {ni − ni−1 | 2 ≤ i ≤ k},
and the delta set of M (also called the set of distances of M) by
[
∆(x)
∆(M) =
x∈M \M ×

(see again [14, Chapter 1.4]). Computations of delta sets in various types of monoids
can be found in [2]-[4].
A monoid M is called a Krull monoid if there is a monoid homomorphism ϕ : M → D
where D is a free abelian monoid and ϕ satisfies the following two conditions:
(1) if a, b ∈ M and ϕ(a) | ϕ(b) in D, then a | b in M,
(2) for every α ∈ D there exist a1 , . . . , an ∈ M with
α = gcd{ϕ(a1 ), . . . , ϕ(an )}.
Clearly, a monoid M is Krull if and only if the associated reduced monoid is Krull. The
basis elements of D are called the prime divisors of M. The above properties guarantee
that Cl(M) = D/ϕ(M) is an abelian group, which we call the class group of M (see
[14, Section 2.3]). Note that since any Krull monoid is isomorphic to a submonoid
of a free abelian monoid, a Krull monoid is commutative, cancellative, and atomic.
The class of Krull monoids contains many well-studied types of monoids, such as the
multiplicative monoid of a ring of algebraic integers (see [1, 12, 14]).
Let G be an abelian group and F (G) the free abelian monoid on G. The elements
of F (G), which we write in the form
Y
X = g1 . . . gl =
g vg (X) ,
g∈G

are called sequences over G. We set −X = (−g1 ) · · · (−gl ). The exponent vg (X) is the
multiplicity of g in X. The length of X is defined as
X
|X| = l =
vg (X)
g∈G

(note that as F (G) and Z(x) are both free abelian monoids; there is
Q really no redundancy in this notation). For every I ⊆ [1, l], the sequence Y = i∈I gi is called a

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S. T. CHAPMAN, F. GOTTI, AND R. PELAYO

subsequence of X. The subsequences are precisely the divisors of X in the free abelian
monoid F (G). The submonoid


X
vg (X)g = 0
B(G) = X ∈ F (G)
g∈G

is known as the block monoid on G, and its elements are referred to as zero-sum
sequences or blocks over G ([14, Section 2.5] is a good general reference on block
monoids). If S is a subset of G, then the submonoid
B(G, S) = {X ∈ B(G) | vg (X) = 0 if g ∈
/ S}
of B(G) is called the restriction of B(G) to S. Block monoids are important examples
of Krull monoids and their true relevance in the theory of nonunique factorizations lies
in the following result.
Proposition 2.1. [14, Theorem 3.4.10.3] Let M be a Krull monoid with class group
G and let S be the set of classes of G which contain prime divisors. Then
L(M) = L(B(G, S)).
Hence, to understand the arithmetic of lengths of factorizations in a Krull monoid,
one merely needs to understand the factorization theory of block monoids. Thus, while
we state our results in the context of general Krull monoids, Proposition 2.1 allows us
to write the proofs using block monoids.
For our purposes, there are two arithmetic properties of the block monoid B(G),
where G is cyclic of order n ≥ 3, which we will use later:
(1) c(B(G)) = n (see [14, Theorem 6.4.7]);
(2) ∆(B(G)) = {1, . . . , c(B(G)) − 2} (see [14, Theorem 6.7.1.4]).
Thus ∆(B(G)) = {1, . . . , n − 2}. For any finite abelian group G with |G| ≥ 3, the set
∆(B(G)) is an interval whose minimum equals 1 but whose maximum is not known in
general [13, 16]. We will be interested in the following types of subsets of ∆(M).
Definition 2.2. Let M be a commutative cancellative monoid and suppose T is a
nonempty subset of ∆(M). We call T realizable in ∆(M) if there is an element x ∈ M
with ∆(x) = T .
Example 2.3. We illustrate some aspects of the last definition with several examples.
(1) Let M be any Krull monoid M with ∆(M) = {c} for c ∈ N. Clearly any
element x ∈ M with |L(x)| > 1 yields ∆(x) = {c} and hence every subset of
∆(M) is realizable. A large class of Dedekind domains (whose multiplicative
monoids are Krull) with such delta sets are constructed in [9]. Another example
of such a monoid is a primitive numerical monoid whose minimal generating
set forms an arithmetic sequence (see [3, Theorem 3.9]).

ON DELTA SETS AND THEIR REALIZABLE SUBSETS
IN KRULL MONOIDS WITH CYCLIC CLASS GROUPS

5

(2) Let G be any infinite abelian group and S any finite nonempty subset of
{2, 3, 4, . . . }. By a well-known theorem of Kainrath ([17], [14, Section 7.4]),
there is a block B ∈ B(G) with L(B) = S. From this, it easily follows that
∆(B(G)) = N. Moreover, it also easily follows that any finite subset T of N
is realizable in ∆(B(G)). In this example we can actually say more. If H is
a monoid, then recall that a submonoid S ⊂ H is divisor-closed if for for all
a ∈ S and b ∈ H one has that b |H a implies that b ∈ S. Set
∆∗ (H) = {min ∆(S) | S ⊂ H is a divisor-closed submonoid and ∆(S) 6= ∅}.
(3)

(4)

(5)
(6)

It follows from [8] that ∆∗ (B(G)) = N.
In general, there are commutative cancellative monoids M with unrealizable
subsets of ∆(M). For our initial example, we again appeal to numerical monoids.
Let S be the monoid of positive integers under addition generated by 4, 6, and
15 (i.e., S = h4, 6, 15i). By [3, Example 2.6], it follows that ∆(S) = {1, 2, 3}.
By [7, Theorem 1], the sequence of sets {∆(n)}n∈S is eventually periodic, and
hence using the periodic bound in that theorem allows one to check for all realizable subsets of {1, 2, 3} in finite time. Using programming from the GAP
NumericalSgps package [10], one can verify that {1}, {1, 2}, and {1, 3} are the
only realizable subsets of ∆(S).
There are three generated numerical monoids that behave differently than the
two types discussed above. For instance, let S be the numerical monoid generated by 7, 10, and 12 (i.e., S = h7, 10, 12i). By [3, Example 2.5], one
has ∆(S) = {1, 2}. Again, using the GAP programming [10], we find that
∆(34) = {1}, ∆(42) = {2}, and ∆(56) = {1, 2}. Thus, all nonempty subsets of
∆(S) are realizable.
If G is cyclic of order n ≥ 3, then by [12, Corollary 2.3.5], for each 1 ≤ i ≤ n−2,
the set {i} is realizable in ∆(B(G)).
Suppose G is an abelian group and S1 ⊆ S2 are subsets of G. By the properties
of the block monoid, if B ∈ B(G, S1 ), then L(B) is equal in both B(G, S1 )
and B(G, S2 ). Thus, if T is realizable in ∆(B(G, S1 )), then T is realizable in
∆(B(G, S2 )). Elementary examples show that this relationship does not work
conversely.

Our eventual goal is to show that in contrast to B(G) where G is infinite abelian,
not all subsets of ∆(B(G)) are realizable when G is finite cyclic.
3. Main Results
Our first lemma will be vital in the proof of Theorem 3.2.
Lemma 3.1. Let G be a cyclic group of order |G| = n ≥ 3, g ∈ G with ord(g) = n,
V = (−g)g, and W = g n . Let u ∈ B(G) and z, z ′ ∈ Z(u), say
z = W r (−W )s BV q , where r, s > 0, q ≥ 0,

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S. T. CHAPMAN, F. GOTTI, AND R. PELAYO

and B is the product of atoms which are not in the set {W, −W, V }. If |z ′ | − |z| = n − 2
and there are no factorizations of u having length between |z| and |z ′ |, then B = 0t for
some t ∈ N0 .
Proof. Since 0 ∈ B(G) is a prime element and L(u) = v0 (u) + L(0−v0(u) u), we may
assume that 0 ∤ u, and we have to show that B = 1 (i.e., B is the empty block). This is
obvious for n = 3, and hence we suppose that n ≥ 4. Assume, by way of contradiction,
that A is an atom of B(G) dividing B. Since 0 ∤ B, it follows that |A| ≥ 2.
Suppose that |A| = 2, say A = (xg)(−xg) with x ∈ {2, . . . , ⌊n/2⌋}. Then
W A(−W ) = (g x (−xg))((−g)x (xg)V n−x
has a factorization of length 2 + n − x. Since 3 < 2 + n − x < n + 1, we obtain a
factorization of u with length strictly between |z| and |z ′ |, a contradiction.
Suppose now that |A| ≥ 3, say A = (xg)(yg)(wg)A′ with x, y, w ∈ {1, . . . , n − 1}
and A′ ∈ F (G). Then two of the three elements −xg, −yg, −wg are either in the
set C = {g, 2g, . . . , ⌊n/2⌋g} or in the set D = {(⌊n/2⌋ + 1)g, . . . , (n − 1)g}. After
renaming xg, yg and wg and exchanging g and −g if necessary, we may suppose that
−xg = (n − x)g ∈ C and −yg = (n − y)g ∈ C. Then WA is divisible by the product of
two atoms ((xg)g n−x)((yg)g n−y ). If (n − x) + (n − y) = n, then n is even, x = y = n/2,
and (xg)(yg) is a zero-sum subsequence of A, a contradiction. Thus (n−x)+(n−y) < n
and WA is the product of at least three atoms. Since A 6= −W , it follows that AW is
a product of at most n − 1 atoms, and thus we obtain a factorization of u with length
strictly between |z| and |z ′ |, a contradiction.

Lemma 3.1 leads us to our first main result.
Theorem 3.2. Let M be a Krull monoid with cyclic class group G of order n ≥ 3. If
x ∈ M and n − 2 ∈ ∆(x), then ∆(x) = {n − 2}.
Proof. By Proposition 2.1, we need only prove the theorem for B(G, S) where S ⊆ G
is the set of classes which contain prime divisors. By our comment in Example 2.3(6),
it suffices to prove our theorem for S = G (i.e., for B(G)).
Let x ∈ B(G) and z, z ′ ∈ Z(x) be such that |z ′ | − |z| = n − 2 and there are no
factorizations of x with length between |z| and |z ′ |. As in Lemma 3.1 we may suppose
that 0 ∤ x. We shall prove that there exists an element g ∈ G such that z is divisible by
the atoms g n and (−g)n . By a previous observation, we know that the catenary degree
of B(G) is n. Therefore, c(x) ≤ n and so there exists an n-chain z = z0 , z1 , . . . , zk = z ′
of factorizations of x from z to z ′ . We also know, by [14, Lemma 1.6.2], that
|zi+1 | − |zi | ≤ d(zi , zi+1 ) − 2 ≤ n − 2.
Let j be the least index such that |zj+1| > |z|. Then |zj | = |z| and |zj+1 | = |z ′ | because
there are no factorizations of x with length between |z| and |z ′ | and also because
|zi+1 | − |zi | ≤ n − 2. It follows that |zj+1| − |zj | = |z ′ | − |z| = n − 2, and so
d(zj , zj+1) ≥ |zj+1 | − |zj | + 2 = n.

ON DELTA SETS AND THEIR REALIZABLE SUBSETS
IN KRULL MONOIDS WITH CYCLIC CLASS GROUPS

7

Therefore, by redefining, if necessary, z and z ′ as zj and zj+1 respectively, we can
assume that d(z, z ′ ) = n.
Let d be the greatest common divisor of z and z ′ in the factorization monoid Z(B(G)),
and take w and w ′ so that z = dw and z ′ = dw ′ . Notice that |w ′ |−|w| = |z ′ |−|z| = n−2.
As max{|w|, |w ′|} = d(z, z ′ ) = n, we have |w ′ | = n and |w| = 2; note that so far | · |
referred to the length in the factorization monoid Z(B(G)) = F (A(B(G))).
It is well known that U ∈ A(B(G)) implies that |U| ≤ n, and |U| = n if and
only if U = g n for some g ∈ G (again by [14, Theorem 5.1.10]; now | · | refers to
F (G)). Since w consists of only two atoms, say w = U1 U2 with U1 , U2 ∈ A(B(G)), we
obtain |U1 U2 | ≤ 2n. Since w ′ is a product of exactly n atoms, say w ′ = V1 · · · Vn with
V1 , . . . , Vn ∈ A(B(G)), we infer that |V1 | = · · · = |Vn | = 2. Then |U1 U2 | = 2n, and since
w ′ is divisible by an atom of length 2, we see that U1 = W = g n and U2 = −W = (−g)n
for some element g ∈ G of order n.
Now we can write z = W r (−W )s BV q where V = (−g)g and B is not divisible by
W , −W , or V . Since there are no factorizations of x having length between |z| and
|z ′ |, Lemma 3.1 implies that B = 1 ∈ Z(B(G)) and z = W r (−W )s V q . Now it easily
follows that ∆(x) = {n − 2} (see [14, Proposition 4.1.2]).

As an immediate consequence of Theorem 3.2, we have the following result.
Corollary 3.3. Let M be a Krull monoid with cyclic class group G of order n ≥ 3. If
T is a nonempty subset of {1, . . . , n − 2} with n − 2 ∈ T but T 6= {n − 2}, then T is
not realizable in ∆(M).
Example 3.4. Let G be a cyclic group of order |G| = 5 and g ∈ G an element of
order 5. We use the last two results to determine all the realizable sets of ∆(B(G)).
Since ∆(B(G)) = {1, 2, 3}, Theorem 3.2 implies that the only possible realizable sets
are {1}, {2}, {3}, and {1, 2}. The singleton sets are guaranteed by Example 2.3(5).
Let B = g 8(2g)(−g)5. We claim that ∆(B) = {1, 2}. If A is an atom with (2g) | A | B,
we deduce that A = (2g)g 3 or A = (2g)(−g)2. The fact that L(g 5(−g)5 ) = {2, 5}
implies the existence of factorizations of B with lengths 3 and 6 when A = (2g)g 3.
Observe also that having A = (2g)(−g)2 determines uniquely the factorization of B
given by z = ((2g)(−g)2)(g(−g))3(g 5 ) ∈ Z(B). As a result, L(B) = {3, 5, 6} and,
therefore, ∆(B) = {1, 2}. Hence, {1}, {2}, {3}, and {1, 2} determine the complete set
of realizable sets of ∆(B(G)).
Our results to this point have produced delta sets of relatively small size. However,
we now prove that we can have realizable sets of large size provided we choose an
adequate finite cyclic group.
Theorem 3.5. For every m ∈ N there exist a Krull monoid M with finite cyclic class
group and an element x ∈ M such that |∆(x)| ≥ m.

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S. T. CHAPMAN, F. GOTTI, AND R. PELAYO

Proof. Choose
Pk a natural number b0 > m and define b1 , . . . , bm recursively by setting
bk+1 = 2( i=0 bi ) + 2m if k > 0. Let B = b1 · · · bm ∈ F (Z) be the sequence over Z,
σ(B) its sum, and

X
6 I ⊆ {1, . . . , m}
Σ(B) =
bi ∅ =
i∈I

its set of subsequence sums.
Choose a finite cyclic group G of order |G| = n > σ(B) and set M = B(G). Let
g ∈ G with ord(g) = n, and consider the element
x = g 2n−σ(B) (−g)n

m
Y
(bi g) ∈ B(G).
i=1

P
Q
For every subset I ⊆ {1, . . . , m}, the element AI = g n− i∈I bi i∈I (bi g) is an atom of
B(G). Suppose there are subsets I, J ⊆ {1, . . . , m} such that AI AJ | x. Then we get
I ∩ J = ∅, I ⊎ J ⊆ {1, . . . , m}, and

X  
X 
n−
bi + n −
bj = vg (AI AJ ) ≤ vg (x) = 2n − σ(B).
i∈I

j∈J

Therefore I ⊎ J = {1, . . . , m} and x = AI AJ (−g)n . Let z be a factorization of x. Then
there is a subset I ⊆ {1, . . . , m} such that AI | z. If AJ ∤ z for J = {1, . . . , m} \ I, then
Y
P
z = AI
((bj g)(−g)bj )((−g)g)n− j∈J bj
j∈J

and |z| = 1 + |J| + n −

P

bj . Therefore

X
L(x) = {3} ∪ 1 + |J| + n −
bj
j∈J



J ⊆ {1, . . . , m} .

j∈J

In order to point out that there are m distinct elements in ∆(x), let


X
Lk = 1 + |J| + n −
bj J ⊆ {1, . . . , m} with max J = k
j∈J

for each k ∈ {0, . . . , m}, with the usual convention that


X
L0 = 1 + |∅| + n −
bj = 1 + n .
j∈∅

If k ∈ {1, . . . , m}, then
min Lk−1 = 1 + (k − 1) + n −

k−1
X
j=1

bj > 1 + 1 + n − bk = max Lk ,

ON DELTA SETS AND THEIR REALIZABLE SUBSETS
IN KRULL MONOIDS WITH CYCLIC CLASS GROUPS

and so
L(x) = {3} ∪

m
]

9

Lk .

k=1

Pk−1
Hence, for each k ∈ {1, . . . , m}, the element min Lk−1 −max Lk = bk − j=1
bj +(k −2)
is in ∆(x), and the growth condition on the elements b1 , . . . , bm guarantees that these
values are pairwise distinct.

Let G be a finite abelian group with |G| ≥ 3. By Theorem 3.2, cyclic groups have
the following property: If max ∆(B(G)) − 2 ∈ ∆(B) for some B ∈ B(G), then
∆(B) = {max ∆(B(G)) − 2}.
It was recently shown in [15] that elementary 2-groups do share this property, and it
is a challenging problem to characterize all such groups. A further wide open question
is to study
Λ(G) = max{|∆(B)| | B ∈ B(G)}.
If G is cyclic of order n ≥ 13, we have shown that 4 ≤ Λ(G) ≤ n − 3.
4. Acknowledgments
It is a pleasure to thank the referee for valuable suggestions which resulted in an
improvement of the manuscript. The authors were supported by National Science
Foundation grants DMS-1035147 and DMS-1045082 and a supplemental grant from
the National Security Agency.
References
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number theory, Amer. Math. Monthly 118 (2011), 901–920.
[2] P. Baginski, S. T. Chapman and G. J. Schaeffer, On the delta set of a singular arithmetical
congruence monoid, J. Théor. Nombres Bordeaux 20 (2008), 45–59.
[3] C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J.
Algebra Appl. 5 (2006), 695–718.
[4] S. Chang, S. T. Chapman and W. W. Smith, On minimum delta set values in block monoids over
cyclic groups, Ramanujan J. 14 (2007), 155–171.
[5] S. T. Chapman, P. A. Garcı́a-Sánchez and D. Llena, The catenary and tame degree of numerical
semigroups, Forum Math. 21 (2009), 117–129.
[6] S. T. Chapman, P. A. Garcı́a-Sánchez, D. Llena, V. Ponomarenko and J. C. Rosales, The catenary
and tame degree in finitely generated commutative cancellative monoids, Manuscripta Math. 120
(2006), 253–264.
[7] S. T. Chapman, R. Hoyer and N. Kaplan, Delta sets of numerical monoids are eventually periodic,
Aequationes Math. 77 (2009), 273–279.
[8] S. T. Chapman, W. A. Schmid and W. W. Smith, On minimal distances in Krull monoids with
infinite class group, Bull. London Math. Soc. 40 (2008), 613–618.
[9] S. T. Chapman and W. W. Smith, On the HFD, CHFD, and k-HFD properties in Dedekind
domains, Comm. Algebra 20 (1992), 1955–1987.

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Department of Mathematics, Sam Houston State University, Huntsville, TX 77341
E-mail address: scott.chapman@shsu.edu
Department of Mathematics, University of Florida, Gainesville, FL 32611
E-mail address: felixgotti@ufl.edu
Department of Mathematics, University of Hawai’i at Hilo, Hilo, HI 96720
E-mail address: robertop@hawaii.edu

