• 대한수학회보
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• 제61권3호
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• pp.867-873
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• 2024

A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Riemannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra ℝD⋉n, where D is a derivation of n whose restriction to the center of n has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.

• 충청수학회지
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• 제6권1호
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• pp.89-94
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• 1993

The main goal of the present paper is to investigate conditions for the separating ideal of a Banach algebra to be nilpotent.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.137-148
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• 1998

We briefly discuss the Lie groups, it's nilpotency and representations of a nilpotent Lie groups. Dixmier and Kirillov proved that simply connected nilpotent Lie groups over $\mathbb{R}$ are monomial. We reformulate the above result at the Lie algebra level.

• Journal of applied mathematics & informatics
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• 제27권5_6호
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• pp.1489-1492
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• 2009

Nagata[3] and Higman[1] showed that nil-algebra of the nilindex n is nilpotent of finite index. In this paper we show that the bounded degree of the nilpotency is less than or equal to $2^n-1$. Our proof needs only some elementary fact about Vandermonde determinant, which is much simpler than Nagata's or Higman's proof.

• 대한수학회논문집
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• 제17권3호
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• pp.451-457
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• 2002

Let (R, m, k) be a maximal commutative k-subalgebra of M$\sub$n/(k) where the index of nilpotency i(m) of m is 3. If the socle of R is of special case, then we can construct some isomorphic maximal commutative subalgebras.

• 대한수학회논문집
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• 제15권1호
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• pp.29-36
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• 2000

The T-ideal of F(X) generated by $x^{n}$ for all x $\in$ X, is generated also by the symmetric polynomials. For each symmetric poly-nomial, there corresponds one row of the incidence matrix. Finding the nilpotency of nil-algebra of nil-index n is equivalent to determining the smallest integer N such that the (n, N)-incidence matrix has rank equal to N!. In this work, we show that the (n, (equation omitted)$^{(1,....,n)}$-incidence matrix is center-symmetric.

• 대한수학회보
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• 제60권1호
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• pp.185-201
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• 2023

With the notion of prime submodule defined by F. Raggi et al. we prove that the intersection of all prime submodules of a Goldie module M is a nilpotent submodule provided that M is retractable and M^(Λ)-projective for every index set Λ. This extends the well known fact that in a left Goldie ring the prime radical is nilpotent.

• 대한수학회보
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• 제48권6호
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• pp.1147-1155
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• 2011

In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup $\mathcal{A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0=1$. Define $\mathcal{A}_{i+1}(G)/\mathcal{A}_i(G)=\mathcal{A}(G/\mathcal{A}_i(G))$ for $i{\geq}1$. By $\mathcal{A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal{A}_{\infty}(G)$, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in $\mathcal{A}(G)$, then G' is also nilpotent.

• Korean Journal of Mathematics
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• 제26권2호
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• pp.175-189
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• 2018

In this paper, at first we generalize the notion of algebra over a field. A ${\Gamma}$-algebra is an algebraic structure consisting of a vector space V, a groupoid ${\Gamma}$ together with a map from $V{\times}{\Gamma}{\times}V$ to V. Then, on every associative ${\Gamma}$-algebra V and for every ${\alpha}{{\in}}{\Gamma}$ we construct an ${\alpha}$-Lie algebra. Also, we discuss some properties about ${\Gamma}$-Lie algebras when V and ${\Gamma}$ are the sets of $m{\times}n$ and $n{\times}m$ matrices over a field F respectively. Finally, we define the notions of ${\alpha}$-derivation, ${\alpha}$-representation, ${\alpha}$-nilpotency and prove Engel theorem in this case.

• 대한수학회보
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• 제53권1호
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• pp.91-102
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• 2016

A chief factor H/K of G is called F-central in G provided $(H/K){\rtimes}(G/C_G(H/K)){\in}{\mathfrak{F}}$. A normal subgroup N of G is said to be ${\pi}{\mathfrak{F}}$-hypercentral in G if either N = 1 or $N{\neq}1$ and every chief factor of G below N of order divisible by at least one prime in ${\pi}$ is $\mathfrak{F}$-central in G. The symbol $Z_{{\pi}{\mathfrak{F}}}(G)$ denotes the ${\pi}{\mathfrak{F}}$-hypercentre of G, that is, the product of all the normal ${\pi}{\mathfrak{F}}$-hypercentral subgroups of G. We say that a subgroup H of G is ${\pi}{\mathfrak{F}}$-embedded in G if there exists a normal subgroup T of G such that HT is s-quasinormal in G and $(H{\cap}T)H_G/H_G{\leq}Z_{{\pi}{\mathfrak{F}}}(G/H_G)$, where $H_G$ is the maximal normal subgroup of G contained in H. In this paper, we use the ${\pi}{\mathfrak{F}}$-embedded subgroups to determine the structures of finite groups. In particular, we give some new characterizations of p-nilpotency and supersolvability of a group.