Cubic rings and their ideals
We give an explicit description of cubic rings over a discrete valuation ring, as well as the description of all ideals of these rings.
Gespeichert in:
| Datum: | 2010 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2880 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508871895810048 |
|---|---|
| author | Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. |
| author_facet | Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. |
| author_sort | Drozd, Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:35Z |
| description | We give an explicit description of cubic rings over a discrete valuation ring, as well as the description of all ideals of these rings. |
| first_indexed | 2026-03-24T02:32:06Z |
| format | Article |
| fulltext |
UDC 512.552+512.715
Yu. A. Drozd (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
R. V. Skuratovskii (Kyiv Nat. Taras Shevchenko Univ.)
CUBIC RINGS AND THEIR IDEALS
КУБIЧНI КIЛЬЦЯ ТА ЇХ IДЕАЛИ
We give an explicit description of cubic rings over a discrete valuation ring, as well as a description of all
ideals of such rings.
Наведено повний опис кубiчних кiлець над дискретно нормованим кiльцем, а також опис усiх iдеалiв
таких кiлець.
Introduction. Ideals of local rings have been studied by a lot of authors from quite
different viewpoints. One of the questions that arise with this respect is on the number of
parameters par(C) defining the ideals of such a ring C up to isomorphism, especially
when it is reduced and of Krull dimension 1. Certainly, it makes sense if the residue field
k is infinite. In [1] it was shown that par(C) = 0, i.e., C has a finite number of ideals (up
to isomorphism), if and only if C is Cohen – Macaulay finite, i.e., has a finite number
of indecomposable non-isomorphic Cohen – Macaulay modules (in the 1-dimensional
reduced case they coincide with torsion free modules). Then Schappert [2] proved that
a plane curve singularity has at most 1-parameter families of ideals if and only if it
dominates one of the strictly unimodal plane curve singularities in the sense of [3], or,
the same, unimodal and bimodal plane curve singularities in the sense of [4]. In [5] this
result was generalized to all curve singularities. Note that this time par(C) = 1 does
not imply that C is Cohen – Macaulay tame, i.e., has at most 1-dimensional families
of indecomposable Cohen – Macaulay modules. Tameness means that C dominates a
singularity of type Tpq [6]. The case par(C) > 1 had not been studied before the second
author described the one branch singularities of type W such that par(C) ≤ 2 [7].
In this paper we study the cubic rings. We describe all such rings, their ideals and,
in particular, establish the value par(C) for any cubic ring C. As a consequence, we
show that a cubic ring is Gorenstein if and only if it is a plane curve singularity (i.e., its
embedding dimension equals 2).
1. Generalities. We denote by D a discrete valuation ring with the ring of fractions
K, the maximal ideal m = tD and the residue field k = D/tD. A cubic ring over D
is, by definition, a D-subalgebra C in a 3-dimensional semisimple K-algebra L, which
is a free D-module of rank 3. We also denote A the integral closure of D in L and
always suppose that A is finitely generated as C-module. Equivalent condition (see, for
instance, [8]): the m-adic completion Ĉ of the ring C has no nilpotent elements. It is
always the case if the algebra L is separable, for instance, if charK = 0. We also set
Am = tmA + D and Jm = tAm−1 = radAm, m > 0.
In what follows, an ideal means a fractional C-ideal in K, i.e., a finitely generated
C-submodule M ⊆ K such that KM = K. Then M is a free D-module of rank 3. We
are going to describe all ideals of cubic rings up to isomorphism. It is known (see, for
instance, [9]) that there is a one-to-one correspondence between C-ideals and Ĉ-ideals,
mapping M to its m-adic completion. This correspondence reflects isomorphisms, i.e.,
maps non-isomorphic ideals to non-isomorphic. So, in what follows we may (and will)
suppose that D is complete with respect to the m-adic topology.
c© YU. A. DROZD, R. V. SKURATOVSKII, 2010
464 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
CUBIC RINGS AND THEIR IDEALS 465
Recall also that the embedding dimension edimC of a local noetherian ring C with
the maximal ideal J and the residue filed k is defined as dimk J/J
2. If C is of Krull
dimension 1 and edimC = 2, C is called a plane curve singularity. In the geometric
case, when C contains a subfield of representatives of k, it actually means that there is a
plane curve C such that C is the completion of the local ring of a singular point x ∈ C.
From the general theory of ramification in finite extensions we see that the following
cases can happen:
One branch, ramified case: L is a field, the maximal ideal of A equals τA,
A/τA ' k and tA = τ3A.
One branch, non-ramified case: L is a field, the maximal ideal of A equals tA
and A/tA = k[θ̄] is a cubic extension of the field k, where θ̄ is a root of an irreducible
cubic polynomial f(x) ∈ k[x].
Two branches, ramified case: L = K1×K, where K1 is a quadratic extension of
K, A = D1 ×D, the maximal ideal of D1 is τD1, D1/τD1 ' k and tD1 = τ2D1.
Two branches, non-ramified case: L = K1×K, where K1 is a quadratic extensi-
on of K, A = D1 × D, the maximal ideal of D1 is tD1 and D1/τD1 = k[θ̄] is a
quadratic extension of the field k,where θ̄ is a root of an irreducible quadratic polynomial
f(x) ∈ k[x].
Three branches case: L = K3, A = D3.
We recall [10, 11] that, for any cubic ring C, every ideal of C is isomorphic either
to an over-ring of C, i.e., a cubic ring B such that C ⊆ B ⊂ L, or to the dual ideal
B∗ = HomD(B,D) of such an over-ring. Hence, to describe all ideals of C, we only
need to describe over-rings of C. Obviously, any cubic ring in L contains some Am.
Therefore, to describe all cubic rings (so their ideals as well), we have to describe the
over-rings of Am. If B is an over-ring of C, they also say that B dominates C.
Since the unique (up to isomorphism) A-ideal is A itself, we proceed by induction:
supposing that all over-rings of Am are known, we find all over-rings of Am+1. If C is
an over-ring of Am+1, then B = CAm is an over-ring of Am, tB ⊂ C and C/tB is
a k-subalgebra in B/tB. If B ⊇ Am−1, then tB ⊇ Jm, hence, C ⊇ Jm + D = Am.
Therefore, the following procedure gives all over-rings of Am+1 which are not over-rings
of Am:
Procedure.
For every over-ring B of Am, which is not an over-ring of Am−1, calculate B̄ =
= B/tB. Set Ā = (Am + tB)/tB ⊆ B̄.
Find all proper subalgebras S ⊂ B̄ such that ĀS = B̄. Equivalently, the natural map
S→ B/BJm must be surjective.
For each such S take its preimage in B.
2. Calculations. 2.1. One branch, ramified case. We set
C2r(α) = D + trαD + t2rA, where v(α) = 1,
C2r+1(α) = D + trαD + t2r+1A, where v(α) = 2,
where v is the discrete valuation related to the ring A, i.e., v(α) = k means that
α ∈ τkD \ τk+1D. Note that C0(α) = A. Obviously, α can be uniquely chosen as
τ + aτ2 for C2r and as τ2 + atτ for C2r+1, where a ∈ D is defined modulo tr.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
466 YU. A. DROZD, R. V. SKURATOVSKII
Theorem 2.1. Every over-ring of Am coincides with tkCr(α) + D for some k, r
such that r+ k ≤ m and some α. The rings Cr(α) are just all plane curve singularities
in this case.
Proof. For m = 1 it is easy and known [1, 12]. So, we use the Procedure for
m > 1, setting B = tkCr(α) + D, where k + r = m. Then the basis of B̄ consists
of the classes of the elements {1, thα, tmτs}, where h = k + [r/2], s ∈ {1, 2} and
s ≡ r (mod 2). Since thα /∈ Jm, the subalgebra S necessarily contains the class of
thα+ ctmτs for some c ∈ D. If k = 0, then m = r and v(tmτs) = 2v(thα). Therefore,
B̄ has no proper subalgebra containing the class of thα+ ctmτs. If k > 0, the preimage
of S is D + (thα + ctmτs)D + tm+1A. It coincides with tk−1Cr+2(α′) + D where
α′ = α+ ctm−hτs.
Now one easily checks that edimCr(α) = 2, while edimC = 3 for all other rings.
The theorem is proved.
2.2. One branch, non-ramified case. We set Cr(α) = D+ trαD+ t2rA0, where
α ∈ A× \ D. Again C0(α) = A0. Note that α can be uniquely chosen as θ + aθ2,
where θ is a fixed preimage of θ̄ in D1 and a ∈ D is defined modulo tr.
Theorem 2.2. Every over-ring of Am coincides with tkCr(α) + D for some k, r
and α with 2r + k ≤ m. The rings Cr(α) are just all plane curve singularities in this
case.
Proof. For m = 1 it is obvious. So, using the Procedure for m > 1, we set
B = tkCr(α)+D with 2r+k = m. Then a basis of B̄ consists of the classes of elements
{1, tr+kα, tmα2} for some α2 ∈ A×\(D+αD). Since tr+kα /∈ Jm, S must contain the
class of an element tr+kα′ = tr+kα+ctmα2 for some c ∈ D. As above, it is impossible
if k = 0. If k > 0, then the preimage of S is D+tr+kα′+tm+1A = tk−1Cr+1(α′)+D.
Now one easily checks that edimCr(α) = 2, while edimC = 3 for all other rings.
The theorem is proved.
2.3. Two branches, ramified case. We denote by v the valuation defined by the
ring D1, by e the idempotent in A such that eA = D1 and set
Cl,q(α) = D + tl(e+ tqα)D + trA, where r = 2l + q,
Cr(α) = D + trαD + t2r+1A.
In both cases α ∈ D1 and v(α) = 1, where v is the valuation defined by the ring D1.
Obviously, α can be uniquely chosen as aτ, where a ∈ D is defined modulo r. Note that
C0,q(α) = D+eD+tqA are just all decomposable rings in this case and C0,0(α) = A.
Theorem 2.3. Every over-ring of Am coincides with either tkCl,r(α) + D or
tkCr(α) +D, where k+ r ≤ m. The rings Cl,q(α) and Cr(α) are just all plane curve
singularities in this case.
Proof. The case m = 1 is obvious. So, using the Procedure, we suppose that
m > 1 and k + r = m. If B = tkCl,q(α) + D, a basis of B̄ consists of the classes
of
{
1, tk+l(e + tqα), tmτ
}
. Since tk+l(e + tqα) /∈ Jm, the subalgebra S must contain
the classe of tk+l(e + tqα′) for some α′ ∈ D1 with v(α′) = 1. Again the case k = 0
is impossible. If k > 0, the preimage of S coincides with tk−1Cl+1,q + D. If B =
= tkCr(α) + D, the calculations are quite similar.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
CUBIC RINGS AND THEIR IDEALS 467
Now one easily checks that edimCl,q(α) = edimCr(α) = 2, while edimC = 3
for all other rings.
The theorem is proved.
2.4. Two branches, non-ramified case. We set
Cl,q(α) = D + tl(e1 + tqα)D + trA,
where r = 2l + q, and α ∈ D1 \ (e1D + tD).
Then α can be chosen as aθ, where θ is a fixed preimage of θ̄ in D1 and a ∈ D is
uniquely defined modulo tl. Again C0,q(α) = D+e1D+ tqA are just all decomposable
rings in this case. Especially, C0,0(α) = A.
Theorem 2.4. Every over-ring of Am coincides with one of the rings tkCl,q(α) +
+ D, where k + r ≤ m. The rings Cl,q(α) are just all plane curve singularities in this
case.
We omit the proof in this case, since it practically repeats the calculations in the
other cases.
2.5. Three branches case. We set
Cl,q(α) = D + tlαD + trA,
where α = e + tqae′, e 6= e′ are primitive idempotent in A, r = 2l + q, a ∈ D×
and a 6≡ 1(mod t) if q = 0. Obviously, a is unique modulo tl. Again C0,q(α) =
= D+ eD+ tqA are just all decomposable rings in this case and C0,0 = A. Note also
that if C = D+ tlαD+ trA, where α = e+ae′ as above with a ≡ 1(mod t), then, for
a ≡ 1(mod tl), C = tlC0,q(1− e− e′) + D, and for a ≡ 1(mod tq) with 0 < q < l,
C = Cl,q(α′) for some α′.
Theorem 2.5. Every over-ring of Am coincides with tkCl,q(α) + D for some α
and some l, q with k + r ≤ m. The rings Cl,q(α) are just all plane curve singularities
in this case.
We also omit the proof in this case, since it practically repeats the calculations in the
other cases.
2.6. Table of plane curve cubic singularities. We present in Table 1 below all plane
curve cubic singularities. In this table s is the number of branches, ∗ marks the unramified
cases (related to the residue field extensions, hence impossible if k is algebraically
closed); x, y are generators of the maximal ideal, v(a) denotes the multivaluation of
an element a ∈ A, i.e., the vector of valuations of its components with respect to the
decomposition of A into the product of discrete valuation rings. The column “type”
shows the correspondence with the Arnold’s classification [4] (§ 15). If chark = 0 and
A is ramified, it actually shows the place of the rings in this classification. If chark = 0
and C is non-ramified, it shows the place of the ring in this classification after the natural
extension of the field k. The validation of this column is given in [5] (Section 2.3). Note
that, following [5], we denote by El,q the singularities Jl,q in the sense of [4]. Such
notations seem more uniform. Note also that the singularities of types E1 and E2 are
actually not cubic, but quadratic, and coincide with those of types A1 and A2 of [4].
Finally, the last column, “par” shows the number of parameters p from the residue field
k which define a unique ring of this type. We will consider this value in the last section.
It does not coincide with the modality in the sense of [4]; the latter equals p− 1.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
468 YU. A. DROZD, R. V. SKURATOVSKII
Table 1
s Name v(x) v(y) Type Par
1 C2r(α) (3) (3r + 1) E6r r
C2r+1(α) (3) (3r + 2) E6r+2 r
1∗ Cr(α) (1) (r) E∗r,0 r
2 Cr(α) (2, 1) (2r + 1,∞) E6r+1 r
Cl,q(α) (2, 1) (2l,∞) El,2q+1 l
2∗ Cl,q(α) (1, 1) (l,∞) E∗l,2q l
3 Cl,q(α) (1, 1, 1) (l, l + q,∞) El,2q l
Remark. The tame cubic plane curve singularities T3,q, q ≥ 6 [6, 13], coincide
with those of types E2,q−6.
3. Ideals. As we have mentioned above, every ideal of a cubic ring C is isomorphic
either to an over-ring B ⊇ C or to its dual B∗ = HomD(B,D). If C is Gorenstein (for
instance, if it is a plane cubic singularity) [14], then C∗ ' C, thus B∗ ' HomC(B,C).
Therefore, to calculate B∗, one has to choose a Gorenstein subring C ⊆ B and to
calculate
HomC(B,C) ' {λ ∈ L |λB ⊆ C} = {λ ∈ C |λB ⊆ C}
(the latter equality holds since 1 ∈ B). This remark easily leads to the following result.
Theorem 3.1. The duals to the cubic rings are as follows:
One branch ramified case: If B = D + tkCr(α), then B∗ ' D + t[r/2]αD +
+ tk+rA.
One branch non-ramified case: If B = D + tkCr(α), then B∗ ' D + trαD +
+ tk+2rA.
Two branches ramified case:
1. If B = D + tkCl,q(α), then B∗ ' D + tl(e+ tqα)D + tk+2l+qA.
2. If B = D + tkCr(α), then B∗ ' D + trαD + tk+2r+1A.
Two branches non-ramified case: If B = D + tkCl,q(α), then B∗ ' D + tl(e+
+ tqα)D + tk+2l+qA.
Three branches case: If B = D + tkCl,q(α), then B∗ ' D + tlαD + tk+2l+qA.
Proof. The proof is immediate if we choose for a Gorenstein subring C ⊆ B the
plane curve singularity C = Ck+r(α) or Ck+l,q(α) depending on the shape of B. For
instance, in two branches ramified case, if B = D + tkCl,q(α) and C = Cl+k,q(α),
then
B∗ ' {λ ∈ C |λB ⊆ C} = tkD + tk+l(e+ tqα)D + t2k+2l+qA '
' D + tl(e+ tqα)D + tk+2l+qA.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
CUBIC RINGS AND THEIR IDEALS 469
Corollary 3.1. If a cubic ring is Gorenstein, it is a plane curve singularity.
Note that it is no more the case for the extensions of bigger degrees. For instance, the
rings Ppq from [6], which are quartic, are Gorenstein (they are complete intersections)
but of embedding dimension 3.
4. Geometric case. Number of parameters. In this section we suppose that our
rings are of geometric nature, i.e., D = k[[t]], where k is algebraically closed. Then one
can consider the number of parameters par(C) defining C-ideals (see [13], Section 2.2,
or [15], Section 3, where it is denoted by par(1;C,A)). Actually, it coincides with
the minimal possible number p for which there is a finite set of families of ideals Ik,
1 ≤ k ≤ m, of dimensions at most p such that every C-ideal is isomorphic to one
belonging to some family Ik. Equivalently, it is the maximal possible p such that there
is a p-dimensional family of ideals I where every isomorphism class of ideals only
occurs finitely many times. In [5] a criterion was established in order that par(C) ≤ 1.
For cubic rings it means that C dominates a singularity of type Em, 18 ≤ m ≤ 20,
or E3,i. The following results give the exact value of par(C) for all cubic rings of
geometric nature. (Note that no unramified case can occur for such rings.)
Theorem 4.1. If C is a cubic ring of geometric nature, par(C) ≤ n if and only if
C dominates one of the singularities of type E12n+i, 6 ≤ i ≤ 8, or E2n+1,q, q ≥ 0.
Proof. Certainly, we have to prove that
(1) every ring of one of the listed types have at most n-parameter families of ideals;
(2) if C dominates no ring of the listed types, it has (n+ 1)-parameter families of
ideals.
Since the calculations in all cases are similar, we only consider the one branch
ramified case. Note first that the rings C2r(α) as well as C2r+1(α) form a r-parametric
family. Indeed, we can choose in the first case α = τ + aτ2, and in the second one
α = τ2+aτ4, where a ∈ D is defined modulo tr, and such a presentation is unique. The
same is true also for tkC2r(α)+D and tkC2r+1(α)+D for any k. Since C2r(α) ⊇ A2r
for all α, we get par(A2r) ≥ r.
Let C dominate neither a ring of type E12n+6, i.e., C4n+2(α), nor a ring of type
E12n+8, i.e., C4n+3(α). Then it contains no element of valuation smaller than 6n+ 6,
so C ⊆ A2n+2. Hence, par(C) ≥ n+ 1.
On the other hand, consider the ring C2r+q(α), where q ∈ {0, 1}. Its over-rings
are of the kind D + tkC2m+q(β), where k + m ≤ r and k + 2m ≤ 2r. Moreover, let
α = τ q+1 + aτ2q+2 and β = τ q+1 + bτ2q+2. Then b is defined modulo tm and b ≡ a
(mod tr−m−k). Therefore, the over-rings with the fixed m, k form a p-parameter family,
where p = min(m, r−m− k). Hence, 2p ≤ r and p ≤ [r/2]. If we set r = 2n+ 1, we
get that par(C4n+2(α)) ≤ n and par(C4n+3(α)) ≤ n for all possible α. It accomplishes
the proof.
Obvious considerations give the number of parameters for special rings.
Corollary 4.1.
par(Cr(α)) = [r/2],
par(Cl,q(α)) = [l/2],
par(Am) = [m/2].
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
470 YU. A. DROZD, R. V. SKURATOVSKII
1. Drozd Y. A., Roiter A. V. Commutative rings with a finite number of integral indecomposable representati-
ons // Izvestia Akad. Nauk SSSR. Ser. mat. – 1967. – 31. – S. 783 – 798.
2. Schappert A. A characterization of strictly unimodal plane curve singularities // Lect. Notes Math. –
1987. – 1273. – P. 168 – 177.
3. Wall C. T. C. Classification of unimodal isolated singularities of complete intersections // Proc. Symp.
Pure Math. – 1983. – 40, № 2. – P. 625 – 640.
4. Arnold V. I., Varchenko A. N., Gusein-Zade S. M. Singularities of differentiable maps. – Moscow: Nauka,
1982. – Vol. 1.
5. Drozd Y. A., Greuel G.-M. On Schappert characterization of unimodal plane curve singularities //
Singularities: The Brieskorn Anniversary Volume. – Birkhäuser, 1998. – P. 3 – 26.
6. Drozd Y. A., Greuel G.-M. Cohen – Macaulay module type // Compos. math. – 1993. – 89. – P. 315 – 338.
7. Skuratovskii R. V. Ideals of one-branched singularities of curves of type W // Ukr. Mat. Zh. – 2009. –
61, № 9. – P. 1257 – 1266.
8. Drozd Y. A. On the existence of maximal orders // Mat. Zametki. – 1985. – 37. – S. 313 – 315.
9. Faddeev D. K. Introduction to multiplicative theory of modules of integral representations // Trudy Mat.
Inst. Steklova. – 1965. – 80. – S. 145 – 182.
10. Faddeev D. K. On the theory of cubic Z-rings // Ibid. – 1965. – 80. – S. 183 – 187.
11. Drozd Y. A. Ideals of commutative rings // Mat. Sbornik. – 1976. – 101. – S. 334 – 348.
12. Jacobinski H. Sur les ordres commutatifs avec un nombre fini de réseaux indécomposables // Acta Math.
– 1967. – 118. – S. 1 – 31.
13. Drozd Y. A. Cohen – Macaulay modules over Cohen – Macaulay algebras // CMS Conf. Proc. – 1996. –
19. – P. 25 – 53.
14. Bass H. On the ubiquity of Gorenstein rings // Math. Z. – 1963. – 82. – S. 8 – 28.
15. Drozd Y. A., Greuel G.-M. Semi-continuity for Cohen – Macaulay modules // Math. Ann. – 1996. – 306.
– P. 371 – 389.
Received 04.01.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
|
| id | umjimathkievua-article-2880 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:32:06Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ee/f8f07c52f3e8758da99e4bf01fe7f4ee.pdf |
| spelling | umjimathkievua-article-28802020-03-18T19:39:35Z Cubic rings and their ideals Кубічні кільця та їх ідеали Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. We give an explicit description of cubic rings over a discrete valuation ring, as well as the description of all ideals of these rings. Наведено повний опис кубічних кілець над дискретно нормованим кільцем, а також опис усіх ідеалів таких кілець. Institute of Mathematics, NAS of Ukraine 2010-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2880 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 4 (2010); 464–470 Український математичний журнал; Том 62 № 4 (2010); 464–470 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2880/2511 https://umj.imath.kiev.ua/index.php/umj/article/view/2880/2512 Copyright (c) 2010 Drozd Yu. A.; Skuratovskii R. V. |
| spellingShingle | Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. Cubic rings and their ideals |
| title | Cubic rings and their ideals |
| title_alt | Кубічні кільця та їх ідеали |
| title_full | Cubic rings and their ideals |
| title_fullStr | Cubic rings and their ideals |
| title_full_unstemmed | Cubic rings and their ideals |
| title_short | Cubic rings and their ideals |
| title_sort | cubic rings and their ideals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2880 |
| work_keys_str_mv | AT drozdyua cubicringsandtheirideals AT skuratovskiirv cubicringsandtheirideals AT drozdûa cubicringsandtheirideals AT skuratovsʹkijrv cubicringsandtheirideals AT drozdyua kubíčníkílʹcâtaíhídeali AT skuratovskiirv kubíčníkílʹcâtaíhídeali AT drozdûa kubíčníkílʹcâtaíhídeali AT skuratovsʹkijrv kubíčníkílʹcâtaíhídeali |