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Let k be a field of characteristic zero and R a factorial affine k-domain. Let B be an affineR-domain. In terms of locally nilpotent derivations, we give criteria for B to be R-isomorphic to the residue ring of a polynomial ring R...
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Let k be a field of characteristic zero and R a factorial affine k-domain. Let B be an affineR-domain. In terms of locally nilpotent derivations, we give criteria for B to be R-isomorphic to the residue ring of a polynomial ring R[X_1, X_2, Y] over R by the ideal (X_1X_2-φ(Y)) for φ(Y)∈R[Y]{set minus}R.
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The main purpose of this paper is to define new concepts on the soft subgroup and soft group, which are called second type nilpotent soft subgroup and nilpotent soft group. Then we study their properties and in sequel, we provide ...
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The main purpose of this paper is to define new concepts on the soft subgroup and soft group, which are called second type nilpotent soft subgroup and nilpotent soft group. Then we study their properties and in sequel, we provide a necessary and sufficient condition for a soft subgroup to be second type nilpotent. Furthermore, we present some relations between a nilpotent group and a second type nilpotent soft subgroup and investigate their properties.
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In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization o...
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In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization of abstract nilpotent groups. For this, we show that a group G is nilpotent if and only if any fuzzy subgroup of G is a g-nilpotent fuzzy subgroup of G . In particular, we construct a nilpotent group via a g-nilpotent fuzzy subgroup. Finally, we characterize the elements of any maximal normal abelian subgroup by using a g-nilpotent fuzzy subgroup.
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Let L be a finitely generated nilpotent Lie algebra over a field K and let d be the smallest integer such that L can be generated by d elements. Let n >= d be a positive integer and suppose that every proper subalgebra of L has cl...
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Let L be a finitely generated nilpotent Lie algebra over a field K and let d be the smallest integer such that L can be generated by d elements. Let n >= d be a positive integer and suppose that every proper subalgebra of L has class at most n. It is not difficult to show that the class of L is at most n + q where q = [n/(d - 1)]. Our main result shows that there exist such Lie algebras of class (exactly) n + q whenever q >= 3 and K has characteristic 0 or prime characteristic p such that p does not divide (q(2) - 1)q/2. (c) 2021 Elsevier Inc. All rights reserved.
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In this paper, by establishing an explicit and combinatorial description of the centralizer of a distinguished nilpotent pair in a classical simple Lie algebra, we solve in the classical case Panyushev's Conjecture, which says tha...
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In this paper, by establishing an explicit and combinatorial description of the centralizer of a distinguished nilpotent pair in a classical simple Lie algebra, we solve in the classical case Panyushev's Conjecture, which says that distinguished nilpotent pairs are wonderful, and the classification problem on almost principal nilpotent pairs. More precisely, we show that distinguished nilpotent pairs are wonderful in types A, B and C, but they are not always wonderful in type D. Also, as the corollary of the classification of almost principal nilpotent pairs, we have that almost principal nilpotent pairs do not exist in the simply-laced case and that the centralizer of an almost principal nilpotent pair in a classical simple Lie algebra is always abelian. (C) 2002 Elsevier Science (USA). All rights reserved. [References: 4]
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Let F be an algebraically closed field of positive characteristic p > 3, and u the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F[T] by T-pn. Let g be the sp...
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Let F be an algebraically closed field of positive characteristic p > 3, and u the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F[T] by T-pn. Let g be the special derivation algebra over 21 which is a simple Lie algebra, and additionally non-restricted as long as n > 1. Let N be the nilpotent cone of g, and G = Aut(g), the automorphism group of g. In contrast with only finitely many nilpotent orbits in a classical simple Lie algebra, there are infinitely many nilpotent orbits in g. In this paper, we parameterize all nilpotent orbits, and obtain their dimensions. Furthermore, the nilpotent cone N is proven to be reducible and not normal. There are two irreducible components in N. The dimension of N is determined. (C) 2016 Elsevier Inc. All rights reserved.
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A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this co...
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A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this condition puts a bound on the class of the Lie algebra. Some p-group results and proofs carry over directly to the Lie algebra case, some carry over with modified proofs and some fail. For the final of these cases, a certain metabelian Lie algebra is constructed to show a case when the p-groups and Lie algebra cases differ.
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In this paper, we have constituted 3-step Fibonacci sequences by the three generating elements of a group of exponent p (p is prime) and nilpotency class n. (C) 2002 Elsevier Inc. All rights reserved. [References: 17]
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We study the simultaneously nilpotent index of a simultaneously nilpotent set of matrices over an antinegative commutative semiring S. We find an upper bound for this index and give some characterizations of the simultaneously nil...
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We study the simultaneously nilpotent index of a simultaneously nilpotent set of matrices over an antinegative commutative semiring S. We find an upper bound for this index and give some characterizations of the simultaneously nilpotent sets when this upper bound is met. In the special case of antinegative semirings with all zero divisors nilpotent, we also find a bound on the simultaneously nilpotent index for all nonmaximal simultaneously nilpotent sets of matrices and establish their cardinalities in case of a finite S. (C) 2016 Elsevier Inc. All rights reserved.
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Given a ring R and polynomials f(x),g(x) ∈ R[x] satisfying f(x)Rg(x) 0, we prove that the ideal generated by products of the coefficients of f (x) and g(x) is nilpotent. This result is generalized, and many well known facts, alon...
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Given a ring R and polynomials f(x),g(x) ∈ R[x] satisfying f(x)Rg(x) 0, we prove that the ideal generated by products of the coefficients of f (x) and g(x) is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if I < R[x] is a left T-nilpotent ideal, then the ideal formed by the coefficients of polynomials in I is also left T-nilpotent.
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