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Journal of the Korean Mathematical Society (03049914)34(4)pp. 949-957
The purpose of this paper is to establish connection between certain complex of modules of generalized fractions and the concept of cosequence in commutative algebra. The main theorem of the paper leads to characterization, in terms of modules of generalized fractions, of regular (co) sequences.
Nagoya Mathematical Journal (21526842)151pp. 37-50
The first part of the paper is concerned, among other things, with a characterization of filter regular sequences in terms of modules of generalized fractions. This characterization leads to a description, in terms of generalized fractions, of the structure of an arbitrary local cohomology module of a finitely generated module over a notherian ring. In the second part of the paper, we establish homomorphisms between the homology modules of a Koszul complex and the homology modules of a certain complex of modules of generalized fractions. Using these homomorphisms, we obtain a characterization of unconditioned strong d-sequences.
Lashkarizadeh Bami M., Abdollahi, A., Woodroofe, R., Woodroofe, R., Zaimi, G., Zaimi, G.
Taiwanese Journal of Mathematics (10275487)(1)pp. 87-95
In the present paper, we shall establish one of our earlier conjectures by proving that on compact subsets of a *-foundation semigroup S with identity and with a locally bounded Borel measurable weight function w, the pointwise convergence and the uniform convergence of a sequence of w-bounded positive definite functions on S which are also continuous at the identity are equivalent.
Let A be a non-zero Artinian R-module. For an arbitrary ideal I of R, we show that the local homology module Hpx(A) is independent of the choice of x whenever 0:A I = 0:A(x1,..., xr). Thus, identifying these modules, we write HpI(A). In this paper we prove that there is a certain duality between HiI(A) and the local cohomology modules and provide some information about the vanishing local homology module HiI(A) which gives an improved form of the main results of [22].
In this note we prove that every infinite group G is 3-abelian (i.e. (ab)3 = a3b3 for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ∈ X and y ∈ Y such that (xy)3 = x3y3.
Communications in Algebra (00927872)27(11)pp. 5633-5638
In this note we show that if G is a finitely generated soluble group, then every infinite subset of G contains two elements generating a nilpotent group of class at most k if and only if G is finite by a group in which every two generator subgroup is nilpotent of class at most k.
Communications in Algebra (00927872)28(6)pp. 2753-2781
We first give a characterization of the core (modulo its center) of an extended affine Lie algebra and then use this characterization to show that as in the case of affine Kac-Moody Lie algebras, many of the known examples of EALAs can be constructed from standard examples by a process known as "twisting".
Bulletin of the Australian Mathematical Society (00049727)62(1)pp. 141-148
Let k be a positive integer. We denote by εk(∞) the class of all groups in which every infinite subset contains two distinct elements cursive Greek chi, y such that [cursive Greek chi,k y] = 1. We say that a group G is an ε*k-group provided that whenever X,Y are infinite subsets of G, there exists cursive Greek chi ∈ X, y ∈ Y such that [cursive Greek chi,k y] = 1. Here we prove that: (1) If G is a finitely generated soluble group, then G ∈ ε3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3. (2) If G is a finitely generated metabelian group, then G ∈ εk(∞) if and only if G/Zk(G) is finite, where Zk(G) is the (k + 1)-th term of the upper central series of G. (3) If G is a finitely generated soluble εk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt(G) is finite. (4) If G is an infinite ε*k-group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.
Bulletin of the Australian Mathematical Society (00049727)64(1)pp. 27-31
We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which every infinite subset contains n mutually disjoint non-empty subsets X1, ..., Xn such that X1 ⋯ Xn ∩ Xσ(1) ⋯ Xσ(n) ≠ 0 for some non-identity permutation σ on the set {1, ..., n}.
Abdollahi, A., Mohammadi hassanabadi a., A.M., Taeri b., B.
Communications in Algebra (00927872)29(4)pp. 1571-1581
Let n > 1 be an integer. A group G is said to be n-rewritable, whenever for any subset {x1, . . ., xn} of elements of G, there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that xτ(1) · · · xτ(n) = xσ(1) · · · xσ(n). In this paper we show that an infinite group G is n-rewritable if and only if for every n infinite subsets X1, . . ., Xn of G there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that Xτ(1) · · · Xτ(n) ∩ Xσ(n) · · · Xσ(n) ≠ 0.
Houston Journal of Mathematics (03621588)27(3)pp. 511-522
Let n and k be positive integers. We say that a group G satisfies the condition ε(n) (respectively, εk(n)) if and only if any set with n + 1 elements of G contains two distinct elements x, y such that [x,t y] = 1 for some positive integer t = t(x,y) (respectively, [x,k y] = 1). Here we study certain groups satisfying these conditions. We prove that if G is a finite group satisfying the condition ε(n), then G is nilpotent if n < 3 and G is soluble if n < 16. If G is a finitely generated soluble group satisfying the condition ε(2), then G is nilpotent. If k and n are positive integers and G is a finitely generated residually finite group satisfying the condition εk(n), then G is nilpotent if n < 3 and G is polycyclic if n < 16. In particular, there is a positive integer c depending only on k such that G/Zc(G) is finite, where Zc(G) is the (c + 1)-th term of the upper central series of G. Also these bounds cannot be improved.
There are two notions of the extended affine root systems in the literature which both are introduced axiomatically. One, extended affine root system (SAERS for short), consists only of nonisotropic roots, while the other, extended affine root system (EARS for short), contains certain isotropic roots too. We show that there is a one to one correspondence between (reduced) SEARSs and EARSs. Namely the set of nonisotropic roots of any EARS is a (reduced) SEARS, and conversely, there is a unique way of adding certain isotropic elements to a SEARS to get an EARS. (It is known that some of extended affine root systems are not the root system of any extended affine Lie algebra.).
Proceedings of the American Mathematical Society (10886826)130(10)pp. 2827-2836
For a given positive integer n and a given prime number p, let r = r (n,p) be the integer satisfying pr-1 < n ≤ pr. We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either pr or pr-1 if p is odd. When p = 2 the best upper bound is pr-1 pr or pr+1. In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.
Bulletin of the Belgian Mathematical Society - Simon Stevin (13701444)9(2)pp. 205-215
In this paper, we consider some combinatorial conditions on infinite subsets of groups, and we obtain in terms of these conditions some characterizations of nilpotent-by-finite and finite-by-nilpotent groups on the class of finitely generated soluble groups.
Journal of the Australian Mathematical Society (14467887)75(3)pp. 313-324
Let R be a commutative Noetherian ring with nonzero identity and let M be a finitely generated R-module. In this paper, we prove that if an ideal I of R is generated by a u.s.d-sequence on M then the local cohomology module H Ii(M) is I-cofinite. Furthermore, for any system of ideals φ of R, we study the cofiniteness problem in the context of general local cohomology modules.
Bulletin of the Australian Mathematical Society (00049727)67(1)pp. 115-119
Let c ≥ 0, d ≥ 2 be integers and Nc(d) be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that Nc(d) is locally nilpotent? We prove that if c ≤ 2d + 2d-1 - 3 then the variety Nc(d) is locally nilpotent and we reduce the question of Gupta about the periodic groups in Nc(d) to the prime power exponent groups in this variety.
