• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.107-118
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• 2017

In this paper we study CR-submanifolds of maximal CR-dimension by investigating extrinsic behaviors of integral curves of characteristic vector field on them. Also we consider the notion of ruled CR-submanifold of maximal CR-dimension which is a generalization of that of ruled real hypersurface and find some characterizations of ruled CR-submanifold of maximal CR-dimension concerning extrinsic shapes of integral curves of the characteristic vector field and those of CR-Frenet curves.

• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.331-340
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• 2023

In this paper, the concept of a maximal chain of ideals is introduced. Some properties of such chains are studied. We introduce some other concepts related to a maximal chain of ideals such as the n-maximal ideal, the maximal dimension of a ring S (M. dim(S)), the maximal depth of an ideal K of S (M.d(K)) and maximal height of an ideal K(M.d(K)).

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.983-992
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• 2023

Let A be a ring and 𝓙 = {ideals I of A | J(I) = I}. The Krull dimension of A, written dim A, is the sup of the lengths of chains of prime ideals of A; whereas the dimension of the maximal spectrum, denoted by dim _𝓙A, is the sup of the lengths of chains of prime ideals from 𝓙. Then dim _𝓙A ≤ dim A. In this paper, we will study the dimension of the maximal spectrum of some constructions of rings and we will be interested in the transfer of the property J-Noetherian to ring extensions.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.147-161
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• 2013

In this paper we determine certain class of $n$-dimensional QR-submanifolds of maximal QR-dimension isometrically immersed in a quaternionic space form, that is, a quaternionic K$\ddot{a}$hler manifold of constant Q-sectional curvature under the conditions (3.1) concerning with the second fundamental form and the induced almost contact 3-structure.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.213-225
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• 1997

Let A be a real $m \times n$ matrix. The column rank of A is the dimension of the column space of A and the maximal column rank of A is defined as the maximal number of linearly independent columns of A. It is wekk known that the column rank is the maximal column rank in this situation.

• Journal of the Korean Mathematical Society
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• v.42 no.4
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• pp.655-672
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• 2005

The purpose of this paper is to study n-dimensional QR-submanifolds of maximal QR-dimension isometrically immersed in a quaternionic projective space and to give sufficient conditions in order for such a submanifold to be a tube over a quaternionic invariant submanifold.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.23-32
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• 2004

Let (B, $m_B$, ${\kappa}$) be a maximal commutative ${\kappa}$-subalgebra of a matrix algebra $M_n(\kappa)$. We will construct a maximal commutative ${\kappa}$-subalgebra (R, $m$, ${\kappa}$) of $M_n+3(\kappa)$ from the algebra B such that the algebra R has dimension greater than the dimension of B by 3. Moreover, we will show a $C_i$-construction doesn't imply a $C^3_2$-construction for $i=1,2$.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.137-143
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• 1992

The purpose of this paper is to provide the affirmative solution of the following conjecture due to Davis and Geramita. Conjecture; Let A=R[T] be a polynomial ring in one variable, where R is a regular local ring of dimension d. Then maximal ideals in A are complete intersection. Geramita has proved that the conjecture is true when R is a regular local ring of dimension 2. Whatwadekar has rpoved that conjecture is true when R is a formal power series ring over a field and also when R is a localization of an affine algebra over an infinite perfect field. Nashier also proved that conjecture is true when R is a local ring of D[ $X_{1}$,.., $X_{d-1}$] at the maximal ideal (.pi., $X_{1}$,.., $X_{d-1}$) where (D,(.pi.)) is a discrete valuation ring with infinite residue field. The methods to establish our results are following from Nashier's method. We divide this paper into three sections. In section 1 we state Theorems without proofs which are used in section 2 and 3. In section 2 we prove some lemmas and propositions which are used in proving our results. In section 3 we prove our main theorem.eorem.rem.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.291-299
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• 1999

Let ${\Omega}$ be the set of all commutative $k$-subalgebras of 14 by 14 matrices over a field $k$ whose dimension is 13 and index of Jacobson radical is 3. Then we will find the equivalent condition for a commutative subalgebra to be maximal.

• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.85-94
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• 1998

The purpose of this paper is to give sample characterizations of n-dimensional CR-submanifolds of (n-1) CR-semifolds of (n-1) CR-dimension immersed in a complex projective space $CP^{(n+p)/2}$ with Fubini-Study metric and we study an n-dimensional compact, orientable, minimal CR-submanifold of (n-1) CR-dimension in $CP^{(n+p)/2}$.