Fields Institute Communications Volume 00, 0000

Prime segments for cones and rings

H.H. Brungs

Department of Mathematical Sciences University of Alberta Edmonton, Alberta Canada T6G 2G1

                      hbrungs@math.ualberta.ca

H. Marubayashi

Department of Mathematics Naruto University of Education Naruto, Japan

                       marubaya@naruto-u.ac.jp

E. Osmanagic

Department of Mathematical Sciences University of Alberta Edmonton, Alberta Canada T6G 2G1

                       eosman@math.ualberta.ca

Abstract. There exists an analogy between the structure of ideals of a cone in a right-ordered group and the structure of ideals of a Dubrovin valuation ring in a simple artinian ring. The structure of ideals of rank one cones and rank one Dubrovin valuation rings can be described completely. One sided versions of this problem are considered in [5] where the ideal theory of right cones is developed and a classification of prime segments is given. The description of the two-sided ideals in this case mirrors the results for cones and Dubrovin valuation rings. However, it remains open whether additional ideals can occur in one particular situation.

1 Prime segments in cones of a right-ordered group

Let G be a group with a positive cone P, i.e., P P ⊆ P, P ∪ P^{−1} = G. If in addition P ∩P^{−1} = {e}, the cone P is called a principal cone of G. The right order “ ≤_{r} ” defined by a ≤_{r} b if and only if ba^{−1} ∈ P for a, b ∈ G is also a left order if and only if aP a^{−1} = P for all a ∈ G.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 16W60, 06F05, 06F15; Secondary 16L30, 16K40, 20M12.

The first author is supported in part by NSERC.

A non-empty subset I of G is called a right P -ideal if IP ⊆ I and I ⊆ aP for some a ∈ G. A left P -ideal and P -ideals are defined similarly. If I ⊆ P then we omit the prefix “P -” and call I a right (left) ideal of P. Moreover, if I ≠ P, then we say that the (right or left) ideal I is proper. A proper ideal I of the cone P is said to be prime (completely prime) if for a, b ∈ P we have a ∈ I or b ∈ I whenever aP b ⊆ I (respectively, ab ∈ I).

A subset H ⊂ G is called convex if x, y ∈ H, g ∈ G, x ≤ g ≤ y implies g ∈ H. The set Σ of all convex subgroups of G is a chain containing {e} and G. If C_{λ}, (λ ∈ Λ) is in Σ, then ∩C_{λ} and ∪C_{λ} are again in Σ. Hence, every element g ≠ e in G defines two subgroups D, C ∈ Σ, so that C ⊃ D, g ∈ C \ D, and Σ contains no convex subgroups strictly between C and D. Such a pair C ⊃ D is called a jump in Σ ([11] and [9]) and it is denoted by D ≺ C.

In this situation there exists a 1 − 1 correspondence between Σ and the set \overline{spec}P of all completely prime ideals of the cone P. More precisely:

Theorem 1.1 Let (G, P) be a right ordered group.

If A ∈ Σ, A ≠ G is a convex subgroup of G then I = P \(P ∩A) is a completely prime ideal of the cone P.

Conversely, if I ≠ P is a completely prime ideal of the cone P, then A = (P \ I) ∪ (P \ I)^{−1} is a convex subgroup of G.

By this correspondence, jumps in Σ correspond to the pairs Q_{1} ⊃ Q_{2} of distinct completely prime ideals of P such that there are no further completely prime ideals between Q_{1} and Q_{2}. Such pairs are called prime segments in P. We also include the possibility that Q_{1} is a minimal completely prime ideal of P and set Q_{2} = ∅ in this case.

Throughout this section P is a cone in a group G. The set of right (left) P -ideals is totally ordered with respect to inclusion. The set U(P) of elements invertible in the cone P is P ∩ P^{−1} and it is a subgroup of G. In particular, P is a principal cone if and only if U(P) = {e}. The subset J(P) of elements not invertible in P is a two-sided ideal called the radical of P; it is the maximal proper right (left) ideal of P. This ideal is completely prime and P = U(P) ∪ J(P), U(P) ∩ J(P) = ∅. We say that the cone P is of rank one if J(P) is the only completely prime ideal of P.

Completely prime ideals of a cone P in a group have the following properties ([1]):

Lemma 1.2 Let P be a cone in a group G. Then:

a): An ideal I in P is completely prime if and only if x^{2} ∈ I implies x ∈ I for

x ∈ P; b): An idempotent ideal I of P is completely prime; c): ∩I^{n} = K is a completely prime ideal for any ideal I of P or K = ∅.

The prime segments of a cone fall into one of the following three categories ([1]):

Theorem 1.3 For a prime segment Q_{1} ⊃ Q_{2} of the cone P in a group G exactly one of the following possibilities occurs:

a): The prime segment Q_{1} ⊃ Q_{2} is invariant, i.e., aQ_{1} = Q_{1}a for all a ∈

Q_{1} \ Q_{2};

b): The prime segment Q_{1} ⊃ Q_{2} is simple, i.e., there are no further ideals of

P between Q_{1} and Q_{2}; c): The prime segment Q_{1} ⊃ Q_{2} is exceptional, i.e., there exists a prime

ideal Q of P that is not completely prime, Q_{1} ⊃ Q ⊃ Q_{2} and then there are no ideals of P between Q_{1} and Q; in addition, ∩Q^{n} = Q_{2}.

Only invariant prime segments occur for cones P in a group G that define an order on G.

For any right P -ideal I we define I^{∗} := ∩hP, hP ⊇ I, the divisorial closure of I. I^{∗} is again a right P -ideal. A right P -ideal I is called divisorial if I = I^{∗}. With D(P) we denote the set of all divisorial P -ideals and on D(P) we define the following operation: I_{1} ∘ I_{2} = (I_{1}I_{2})^{∗} for all I_{1}, I_{2} ∈ D(P). With H(P) we denote the set of all invariant principal P -ideals, that is zP = P z for some z ∈ G.

The following result is essential for describing the ideals of a rank one cone ([1]).

Theorem 1.4 Let P be a cone of rank one in the group G. Then: a): (D(P), ∘) is a linearly ordered group; b): (H(P), ∘) is a subgroup of D(P); c): The inverse of the divisorial P -ideal I in D(P) is I^{−1} := {g ∈ G | IgI ⊆ I}.

Using the fact that for a cone P of rank one, (D(P), ∘) is a group, the structure of two-sided ideals of P in this case can be described completely ([1]).

Theorem 1.5 Let P be a rank one cone in a group G and J = J(P). Then exactly one of the following possibilities occurs:

a): The cone P is invariant, i.e., aP = P a for all a ∈ P and either

i): J = zP = P z for some z ∈ P and then D(P) ≅ ⟨J⟩ ≅ H(P) is an

infinite cyclic group, or ii): D(P) ≅ (ℝ, +) and H(P) is a dense subgroup of D(P);

b): The cone P is nearly simple, i.e., J is the only proper ideal of P and then

D(P) = H(P) = {P } is the trivial group; c): The cone P is exceptional, i.e., there exists a prime ideal Q that is not

completely prime with P ⊃ J ⊃ Q, no further ideals exist between J and Q and ∩Q^{n} = ∅. Then D(P) = ⟨Q⟩ is the infinite cyclic group generated by Q = Q^{∗} and an integer k ≥ 0 exists with H(P) = ⟨(Q^{k})^{∗}⟩.

The Case c) splits into the following subcases: c_{0}) c_{0}) c_{0}): If k = 0, then P has no principal ideals different from P, i.e., H(P) = {P },

and

P ⊃ J ⊃ Q ⊃ Q^{2} ⊃ ⋯

is the chain of all ideals of P. c_{1}) c_{1}) c_{1}): If k = 1 then D(P) = ⟨Q⟩ = H(P), i.e., the ideal Q is principal, Q =

zP = P z and

P ⊃ J ⊃ zP ⊃ zJ ⊃ z^{2}P ⊃ z^{2}J ⊃ ⋯

is the chain of all ideals of P. c_{k}) c_{k}) c_{k}): If k > 1 then (Q^{k})^{∗} = zP = P z, Q^{k} = zJ, and (Q^{k})^{m} = z^{m}J, for m ≥ 1.

Hence, the chain of all ideals of P is:

P ⊃ J ⊃ Q ⊃ Q^{2} ⊃ ⋯ ⊃ Q^{k−1} ⊃ zP ⊃ zJ = Q^{k} ⊃ Q^{k+1} ⊃ ⋯ ⊃ Q^{2k−1} ⊃

z^{2}P ⊃ z^{2}J = Q^{2k} ⊃ ⋯

Note that a subring R of a skew field D is a chain domain if and only if the semigroup (R\{0}, ·) is a cone in the multiplicative group of D. This correspondence preserves the properties to be a proper ideal, a prime ideal and a completely prime ideal. Hence, Theorem 1.5 applies to chain domains as well. Several related results have been proved earlier ([4], [3], [8]). Also, all cases in Theorem 1.5 can be realized, that is, examples for each case have been constructed ([3], [8] and [7]).

2 Prime segments in a Dubrovin valuation ring of a simple artinian ring

In 1984 N. Dubrovin introduced a new class of noncommutative valuation rings in a simple artinian ring, now called Dubrovin valuation rings. A very well developed ideal theory and extension theory in the finite dimensional case justify the name valuation rings.

Definition 2.1 A subring R in a simple artinian ring Q is called Dubrovin valuation ring of Q if there exists an ideal M of R such that

i): R/M is a simple artinian;

ii): For all q ∈ Q \ R there exist r, r′ ∈ R with qr, r′q ∈ R \ M.

The ideal M is then the Jacobson radical J(R) of R and M = J(R) is a unique maximal ideal of R. If R/J(R) is a skew field, then Q is a skew field and R is a total valuation ring of Q. In the case that Q = K is a field the classes of Krull valuation rings in K, total valuation rings in K and Dubrovin valuation rings all agree.

Examples of Dubrovin valuation rings include chain domains and matrix rings over chain domains. A particular example is the ring of quaternions over the p-adic valuation ring ℤ_{(p)} with p an odd prime, in the ring Q of rational quaternions.

Let R be a Dubrovin valuation ring of Q. Then the following holds ([6] and [10]):

(D1): Every finitely generated one-sided ideal of R is principal.

(D2): R is a Bezout order in Q such that R/J(R) is a simple artinian.

(D3): The two-sided ideals of R are totally ordered by inclusion.

(D4): Every overring S of R in Q is again a Dubrovin valuation ring of Q and P = J(S) is a prime ideal of R such that R/P is a Goldie. Conversely , if P is a prime ideal of R such that R/P is a Goldie, then the set C_{R}(P) := {r ∈ R | r + P is regular in R/P } is a regular right and left Ore set and the localization R_{P} of R with respect to C_{R}(P) is an overring of R in Q.

Considering the property (D4), the following definition is natural.

Definition 2.2 A prime ideal P of R is called Goldie prime if R/P is a prime Goldie ring.

A prime segment of R consists of two distinct Goldie primes P_{1} ⊃ P_{2} in R so that no further Goldie prime exists between P_{1} and P_{2}.

A Dubrovin valuation ring R is said to be of rank one if {0} and J(R) are the only Goldie prime ideals of R.

Remark 2.3 The ideals {0} and J(R) are Goldie primes and if R is a total valuation ring then a prime ideal P of R is a Goldie prime if and only if P is completely prime.

Goldie prime ideals of a Dubrovin valuation ring R have the following properties ([2]).

Lemma 2.4 Let R be a Dubrovin valuation ring of Q. Then:

a): Any union and intersection of Goldie prime ideals is again Goldie prime; b): An idempotent proper ideal of R is a Goldie prime; c): ∩I^{n} is a Goldie prime for any ideal I ≠ R of R.

The next result shows that there are exactly three types of prime segments in a Dubrovin valuation ring. Property c) in Lemma 2.4 is crucial in the proof of the following classification theorem ([2]).

Theorem 2.5 For a prime segment P_{1} ⊃ P_{2} of a Dubrovin valuation ring R exactly one of the following possibilities occurs:

a): The prime segment P_{1} ⊃ P_{2} is archimedean, i.e., for all a ∈ P_{1} \P_{2} there

exists an ideal I ⊆ P_{1} so that a ∈ I and ∩I^{n} = P_{2}. b): The prime segment P_{1} ⊃ P_{2} is simple, i.e., there are no further ideals

between P_{1} and P_{2}. c): The prime segment P_{1} ⊃ P_{2} is exceptional, i.e., there exists a prime ideal P that is not Goldie prime, P_{1} ⊃ P ⊃ P_{2} and then there are no ideals between P_{1} and P and ∩P^{n} = P_{2}.

Proof For a proof one considers L(P_{1}), the union of ideals I of R properly contained in P_{1}. Three cases may happen.

Case 1.

If L(P_{1}) = P_{2}, the segment P_{1} ⊃ P_{2} is simple.

Case 2.

If P_{1} ⊃ L(P_{1}) ⊃ P_{2} and P_{1} = P_{1}^{2}, then L(P_{1}) is a prime ideal. Hence, the prime segment P_{1} ⊃ P_{2} is exceptional. Note that the converse also holds, i.e., if the prime segment P_{1} ⊃ P_{2} is exceptional, then P_{1} ⊃ L(P_{1}) ⊃ P_{2} and P_{1} = P_{1}^{2}.

Case 3.

If P_{1} ⊃ P_{1}^{2} or P_{1} = L(P_{1}) the prime segment P_{1} ⊃ P_{2} is archimedean.

If R is a Dubrovin valuation ring in the simple artinian ring Q finite dimensional over its center, then R has archimedean prime segments only.

An R− ideal A of Q (fractional ideal) is called divisorial if A = A^{∗} := ∩cS, cS ⊇ A, where c ∈ U(Q) and S = O_{r}(A) := {q ∈ Q | Aq ⊆ A}. With D(R) we denote the set of all divisorial R− ideals and on D(R) we define a partial operation “ ∘ ” by: A ∘ B = (AB)^{∗}, for A, B ∈ D(R) with O_{r}(A) = O_{l}(B). With respect to “ ∘ ”, D(R) becomes an algebraic structure known as Brandt groupoid without connectivity property ([6]).

Let R be a Dubrovin valuation ring of rank one. Then by (D4), Q is the only proper overring of R, i.e., R is a maximal order in Q. In this case the operation “ ∘ ” is defined on the whole D(R). Using this fact, we can obtain the following result essential for describing the two-sided ideals of a Dubrovin valuation ring of rank one ([2]).

Theorem 2.6 Let R be a Dubrovin valuation ring of rank one. Then:

a): (D(R), ∘) is a group that is order isomorphic to a subgroup of (ℝ, +). b): (H(R), ∘), where H(R) := {I ∈ D(R) | 0 ≠ I = aR, 0 ≠ a ∈ I}, is a

subgroup of D(R).

It follows from the next result that if the groups D(R) and H(R) are given, then all ideals of a rank one Dubrovin valuation ring are known ([12]).

Lemma 2.7 Let R be a Dubrovin valuation ring of Q. If A is an R− ideal of Q that is not divisorial, i.e., A ⊂ A^{∗}, then A^{∗} = cS and A = cJ(S), where S = O_{r}(A) and C ∈ U(Q). In particular, if R is of rank one, then A^{∗} = cR and A = cJ(R).

The next result describes all possibilities for the groups D(R) and H(R) if R is a rank one Dubrovin valuation ring ([2]).

Theorem 2.8 Let R be a rank one Dubrovin valuation ring and J = J(R). Then exactly one of the following possibilities occurs:

a): The segment J ⊃ (0) is archimedean and i): J ⊃ J^{2} and then D(R) ≅ ⟨J⟩ ≅ H(R) is an infinite cyclic group; or ii): J = J^{2} and then D(R) ≅ (ℝ, +) and H(R) is a dense subgroup of

D(R). b): The segment J ⊃ (0) is simple and then D(R) = H(R) = {R} is the trivial

group. c): The segment J ⊃ (0) is exceptional and P with J ⊃ P ⊃ (0) is a non Goldie prime in R. Then D(R) = ⟨P ⟩ is the infinite cyclic group generated by P = P^{∗} and an integer k ≥ 0 exists with H(R) = ⟨(P^{k})^{∗}⟩.

Proof To prove this result, one uses Theorem 2.6 to understand the group D(R) and Lemma 2.7 to describe all ideals of R in terms of D(R) and H(R).

The Case c) splits into the following subcases:

c_{0}): If k = 0 then R ⊃ J ⊃ P ⊃ P^{2} ⊃ ⋯ ⊃ (0) is the chain of all ideals of R. c_{1}): If k = 1 then P = P^{∗} = aR = Ra and

R ⊃ J ⊃ aR ⊃ aJ ⊃ a^{2}R ⊃ a^{2}J ⊃ ⋯ ⊃ (0)

is the chain of all ideals of R. c_{k}): If k > 1 then (P^{k})^{∗} = aR = Ra, P^{k} = aJ, and

R ⊃ J ⊃ P ⊃ P^{2} ⊃ ⋯ ⊃ P^{k−1} ⊃ aR ⊃ aJ = P^{k} ⊃ P^{k+1} ⊃ ⋯ ⊃ P^{2k−1} ⊃

a^{2}R ⊃ a^{2}J = P^{2k} ⊃ ⋯ ⊃ (0)

is the chain of all ideals of R.

3 Prime segments in right cones

Let G be a group and C a submonoid of G that generates G, i.e., G = ⟨C⟩. Then C is called a right cone of G if for any a, b ∈ C either aC ⊆ bC or bC ⊆ aC.

One sided ideals, ideals, prime ideals and completely prime ideals can be defined for C as they were defined for cones in Section 1. With U(C) = C ∩ C^{−1} we denote the subgroup of units of C, and with J = J(C) = C \ U(C) the maximal proper right ideal of C which is also an ideal.

A prime segment P_{1} ⊃ P_{2} of C consists of two distinct completely prime ideals of C so that no further completely prime ideal of C lies between P_{1} and P_{2} ; as in Section 1, we call P_{1} ⊃ ∅ a prime segment if P_{1} is a minimal completely prime ideal of C.

For right cones the following two facts are essential in the proof of the classification theorem for prime segments:

Lemma 3.1 Let C be a right cone in a group G. Then:

a): A proper ideal I of C is completely prime if and only if a^{2} ∈ I implies

a ∈ I for a ∈ C; b): ∩I^{n} = K is a completely prime ideal for any ideal I of C or K = ∅.

Theorem 3.2 For a prime segment P_{1} ⊃ P_{2} of a right cone C in a group G exactly one of the following possibilities occurs:

a): The prime segment P_{1} ⊃ P_{2} is right invariant, i.e., P_{1}a ⊆ aP_{1} for all

a ∈ P_{1} \ P_{2}. b): The prime segment P_{1} ⊃ P_{2} is simple, i.e., there are no further ideals of

C between P_{1} and P_{2}. c): The prime segment P_{1} ⊃ P_{2} is exceptional, i.e., there exists a prime ideal Q of C that is not completely prime, P_{1} ⊃ Q ⊃ P_{2} and then there are no ideals of C between P_{1} and Q; in addition, ∩Q^{n} = P_{2}.

We can apply this result to a rank one right cone C, i.e.,where the maximal ideal J = J(C) is the only completely prime ideal.

Theorem 3.3 Let C be a rank one right cone in a group G and J = J(C). Then exactly one of the following possibilities occurs:

a): The cone C is right invariant, i.e., Ca ⊆ aC for all a ∈ C and either

i): J = aC for some a ∈ C and the powers of J are the only ideals of

C, or ii): J is not finitely generated as a right ideal of C; the semigroup of principal right ideals corresponds to a dense subsemigroup of (ℝ^{≥0}, +) and there is an additional ideal in C corresponding to any non-negative real number;

b): The cone C is nearly simple, i.e., J is the only proper ideal of C; c): The cone C is exceptional, i.e., there exists a a prime ideal Q that is not completely prime with C ⊃ J ⊃ Q, no further ideals exist between J and Q and ∩Q^{n} = ∅.

An integral domain R is called a right chain domain if aR ⊆ bR or bR ⊆ aR hold for a, b ∈ R; such a ring has a skew field D of quotients. A subring R of a skew field D is a right chain domain if and only if R \ {0} is a right cone of D \ {0}.

Using this fact, the results in Theorem 3.2 and 3.3 apply to right chain domains as well. However, we are not able to classify all two-sided ideals of exceptional right cones of rank one as was done in Theorem 1.5, c) for cones, and in Theorem 2.8 for rank one Dubrovin valuation rings.

It would be desirable to have a universal theory that covers all three cases discussed in this paper.

References

[1] H.H. Brungs and N.I. Dubrovin, Classification of Chain Rings, preprint.

[2] H.H. Brungs, H. Marubayashi and E. Osmanagic, A Classification of Prime Segments in Simple Artinian Rings, to appear in Proc. Amer. Math. Soc.

[3] H.H. Brungs and M. Schröder, Prime segments of skew fields, Canadian J. Math. 47 (1995), 1148-1176.

[4] H.H. Brungs, and G. Törner, Chain rings and prime ideals, Arch. Math. 27 (1976), 253-260.

[5] H.H. Brungs, and G. Törner, Ideal theory of right cones and associated rings, J. Algebra 210 (1998), 145-164.

[6] N.I. Dubrovin, Noncommutative valuation rings, Trudy Moskov. Mat. Obsch. 45 (1982), 265-280; English transl., Trans. Moscow. Math. Soc. 45 (1984), 273-287.

[7] N.I. Dubrovin, The rational closure of group rings of left orderable groups, Math. Sbornik 187 (7) (1993), 3-48.

[8] T.V. Dubrovin and N.I. Dubrovin, Cones in groups (Russian), Math. Sbornik 187 (7) (1996), 59-74.

[9] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, New York, 1963.

[10] J. Gräter, Prime PI rings in which finitely generated right ideals are principal, Forum Math. 4 (1992), 447-463.

[11] V.M. Kopytov, Right Ordered Groups, J. Wiley, 1998.

[12] H. Marubayashi, H. Miyamoto and A. Ueda, Non-commutative valuation rings and semiheridetary orders, Kluwer Acad. Publ., Dordrecht, 1997.