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1.
V. K. Kharchenko 《Algebra and Logic》1996,35(1):49-58
Let L be a finite-dimensional differential Lie algebra acting on a prime ring R and let the inner part {ie49-1} of L be quasi-Frobenius.
Then a constant ring RL is prime iff {ie49-2} is a differentially simple ring. A ring of constants is semiprime iff {ie49-3} is a direct sum of differentially
simple rings, and the prime dimension of a constant ring is equal to the number of differentially simple summands {ie49-4}.
The Galois closure of L is obtained from L by adding all the inner derivations of a symmetric Martindale quotient ring which
agree with elements from {ie49-5}.
Supported by RFFR grant No. 93-01-16171 and by ISF grant RPS000-RPS300.
Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 88–104, January–February, 1996. 相似文献
2.
A. N. Koryukin 《Algebra and Logic》1997,36(4):236-244
In [1], the question was posed as to whether or not all algebraic relations of skew derivations of prime rings follow from
primitive algebraic relations. Here we argue to obtain a negative answer to a milder question, and namely, an example is constructed
in which a pointed Hopf algebra H (generated as an algebra with unity by its relatively primitive elements) acts trivially
on the generalized centroid C of a prime ring R, but not all algebraic relations of skew derivations (corresponding to relatively
primitive elements in H) follow from primitive algebraic ones. The R in the counterexample is a free associative C-algebra.
Supported by ISF grant No. RPS300 and by RFFR grant No. 95-01-01356a.
Translated from Algebra i Logika, Vol. 36, No. 4, pp. 407–421, July–August, 1997. 相似文献
3.
Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/gk( F) ¢F/\gamma _{k}\left( F\right) ^{^{\prime }}. 相似文献
4.
5.
A class of varieties V (including all finitely based lattice varieties) is determined for which the elementary equivalence
of lattices of subalgebras of free V-algebras, Fv(X) and Fv(Y), is equivalent to sets X and Y being second-order equivalent.
Supported by RFFR grant No. 99-01-00571.
Supported by the National Research Foundation of the Republic of South Africa, and by the University of Cape Town Research
Committee.
Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 595–601, September–October, 2000. 相似文献
6.
In this paper, we determine the X-inner automorphisms of the smash product R # U(L) of a prime ring R by the universal enveloping algebra U(L) of a characteristic 0 Lie algebra L. Specifically, we show that any such automorphism σ stabilizing R can be written as a product σ = σ1σ2, where σ1 is induced by conjugation by a unit of Q3(R), the symmetric Martindale ring of quotients of R, and σ2 is induced by conjugation by a unit of Q3(T). Here S = Ql(R) is the left Martindale ring of quotients of R and T is the centralizer of S in S # U(L) - R # U(L). One of the subtleties of the proof is that we must work in several unrelated overrings of R # U(L). 相似文献
7.
V. K. Kharchenko 《Algebra and Logic》1997,36(2):133-144
Let L be a finite-dimensional restricted differential Lie C-algebra of R-continuous derivations of a prime ring R of characteristic
p>0, with generalized centroid C. We prove that if the associative inner part of L is quasi-Frobenius then R contains a nonzero
element a and elements v1,…,vn, such that for any x∈R we have the expansion
, where
are homorphisms of right RL-modules
. This gives rise to a certain relation on a ring over some subring, known as Shirshov local finiteness. The structure of
(R, RL)-subbimodules in a left Martindale ring of quotients is elucidated.
Supported by RFFR grant No. 95-01-01356a, and by the CONACYT of Mexico, Catedra Patrimonial 940411.
Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 219–238, March–April, 1997. 相似文献
8.
Let R be a reduced ring with Q its Martindale symmetric ring of quotients, and let B be the complete Boolean algebra of all idempotents in C, where C is the extended centroid of R. It is proved that every minimal prime ideal of R must be of the form mQ∩R for some maximal ideal m of B but the converse is in general not true. In addition, if R is centrally closed or has only finitely many minimal prime ideals, then the converse also holds. By applying the explicit expression, many properties of minimal prime ideals of reduced rings are realized more easily. 相似文献
9.
Let R be a ring graded by an abelian group.We study prime ideals of R that are maximal for not containing nonzero homogeneous elements.Also prime ideals of the symmetric graded Martindale ring of quotients of R are investigated.The results are applied to study when R is a Jacobson ring in case R is a Z-graded ring or a group ring of a finitely generated abelian group, or in case R is right Noetherian and strongly graded by a polycyclic-by-finite group. 相似文献
10.
V. T. Markov 《Algebra and Logic》1997,36(5):330-335
It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent.
If, moreover, the R-module M and the left idealR(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a
semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings.
Moreover, first, a right Artinian ring R such that rad(R)2=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful
Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely
generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension.
Supported by RFFR grant No. 26-93-011-1544.
Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997. 相似文献
11.
The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring
of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a.
Partially supported by the Ministerio de Educación y Ciencia and Fondos Feder, jointly, trough projects MTM2004-03845, MTM2007-61978
and MTM2004-06580-C02-02, MTM2007-60333, by the Junta de Andalucía, FQM-264, FQM336 and FQM02467 and by the Plan de Investigación
del Principado de Asturias FICYT-IB05-017. 相似文献
12.
Centers of universal envelopes for Mal’tsev algebras are explored. It is proved that the center of the universal envelope
for a finite-dimensional semisimple Mal’tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite
number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module
over its center. Centers of universal enveloping algebras are computed for some Mal’tsev algebras of small dimensions.
Supported by FAPESP grant No. 04/08537-4 and by SO RAN grant No. 1.9.
Supported by FAPESP grant Nos. 05/60142-7, 05/60337-2 and by CNPq grant No. 304991/2006-6.
__________
Translated from Algebra i Logika, Vol. 46, No. 5, pp. 560–584, September–October, 2007. 相似文献
13.
We present the construction for a u-product G1 ○ G2 of two u-groups G1 and G2, and prove that G1 ○ G2 is also a u-group and that every u-group, which contains G1 and G2 as subgroups and is generated by these, is a homomorphic image of G1 ○ G2. It is stated that if G is a u-group then the coordinate group of an affine space Gn is equal to G ○ Fn, where Fn is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian
group or wreath product of two free Abelian groups of finite ranks.
__________
Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005.
Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education
grant No. E00-1.0-12. 相似文献
14.
An automorphism of a (profinite) group is called normal if each (closed) normal subgroup is left invariant by it. An automorphism
of an abstract group is p-normal if each normal subgroup of p-power, where p is prime, is left invariant. Obviously, the inner
automorphism of a group will be normal and p-normal. For some groups, the converse was stated to be likewise true. N. Romanovskii
and V. Boluts, for instance, established that for free solvable pro-p-groups of derived length 2, there exist normal automorphisms
that are not inner. Let N2 be the variety of nilpotent groups of class 2 and A the variety of Abelian groups. We prove the following results: (1) If
p is a prime number distinct from 2, then the normal automorphism of a free pro-p-group of rank ≥2 in N2A is inner (Theorem 1); (2) If p is a prime number distinct from 2, then the p-normal automorphism of an abstract free N2A-group of rank ≥2 is inner (Theorem 2).
Supported by RFFR grant No. 93-01-01508.
Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 249–267, May–June, 1996. 相似文献
15.
V. G. Bardakov 《Algebra and Logic》2000,39(4):224-251
We study commutator length in free groups. (By a commutator lengthcl(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k
commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F2 (Thm. 1). Moreover, for every element z ε F′2 and for any natural m, the following estimate derives:cl(zm) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute
commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory,
we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of
pairs, and also for its corresponding Lie algebra PL. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that
in a free group F2k with free generators a1, b1, ..., ak, bk, k εN, every natural m satisfiescl(([a1, b1] ... [ak, bk])m)=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely
generated group G is well known. Associated with a derived subgroup of F2 is some series depending on two variables which bears information not only on the number of elements of prescribed length
but also on the number of elements of prescribed commutator length. A number of open questions are formulated.
Supported by RFFR grant No. 98-01-00699.
Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 395–440, July–August, 2000. 相似文献
16.
Matej Brešar 《Algebras and Representation Theory》2016,19(6):1437-1450
The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q m l (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q m s (R), but not for the symmetric Martindale ring of quotients Q s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations. 相似文献
17.
Yuan-Tsung Tsai 《代数通讯》2013,41(10):3608-3615
Let R be a domain and R[X; D] the Ore extension of R by a sequence D of derivations of R. If D has length ≥ 2, we show that the symmetric Utumi quotient ring of R[X; D] is U s (R)[X; D], where U s (R) is the symmetric Utumi quotient ring of R. Consequently, X-inner automorphisms of R[X; D] are induced by units of U s (R) and the extended centroid of R[X; D] consists of those elements α in the center of U s (R) such that δ(α) = 0 for all δ ? D. These extend the known results for free algebras. 相似文献
18.
In this paper we introduce Martindale quotients of Jordan algebras over arbitrary rings of scalars with respect to denominator filters of ideals. For any denominatored algebra, we show the existence of maximal Martindale quotients naturally containing all Martindale quotients of the algebra with respect to the given denominator filter. 相似文献
19.
V. T. Filippov 《Algebra and Logic》1996,35(2):120-128
Let Φ be an associative commutative ring with unity, 1/6 ∈ Φ, write A for a Mal’tsev algebra over Φ, suppose that on A, the
function h(y, z, t, x, x)=2[{yz, t, x}x+{yx, z, x}t], where {x, y, z}=(xy)z−(xz)y+2x(yz), is defined, and assume that H(A)
is a fully invariant ideal of A generated by the function h. The algebra A satisfying an identity h(y, z, x, x, x)=0 [h(y,
z, t, x, x)=0] is called a Mal’tsev h0-algebra (h-algebra). We prove that in any Mal’tsev h0-algebra, the inclusion H(A)·A2⊆Ann A holds withAnnA the annihilator of A. This means that any semiprime h0-algebra A is an h-algebra. Every prime h0-algebra A is a central simple algebra over the quotient field Λ of the center of its algebra of right multiplications, R(A),
and is either a 7-dimensional non-Lie algebra or a 3-dimensional Lie algebra over Λ.
Supported by RFFR grant No. 94-01-00381-a.
Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 214–227, March–April, 1996. 相似文献
20.
Let (B
i
)
i∈I
be a set of Lie algebras; let X be a free Lie algebra; let
* X be their free sum; let R be an ideal of F such that R ⋂ B
i
= 1 (i ∈ I); let V be a variety of Lie algebras such that V(R) is an ideal of F. Under some restrictions, we construct an embedding of F/V(R) into the verbal wreath product of a free algebra of the variety V with F/R.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 235–241, 2004. 相似文献