• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.73-88
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• 2016

We consider ${\mathcal{T}}_{\infty}(F)$ - the space of all innite upper triangular matrices over a eld F. We give a description of all linear maps that satisfy the property: if rank(x) = 1, then $rank({\phi}(x))=1$ for all $x{\in}{\mathcal{T}}_{\infty}(F)$. Moreover, we characterize all injective linear maps on ${\mathcal{T}}_{\infty}(F)$ such that if rank(x) = k, then $rank({\phi}(x))=k$.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.541-547
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• 2008

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with $n\;{\geq}\;3$. We denote by $M_n(R)$ the ring of all $n{\times}n$ matrices over R. Let ($J_n(R)$) be the additive subgroup of $M_n(R)$ generated additively by all idempotent matrices. Let ($D=J_n(R)$) or $M_n(R)$. In this paper, by using an additive idem potence-preserving result obtained by Coo (see [4]), I characterize (i) the additive preservers of tripotence from D to $M_m(R)$ when 2 and 3 are units of R; (ii) the additive preservers of inverses (respectively, Drazin inverses, group inverses, {1}-inverses, {2}-inverses, {1, 2}-inverses) from $M_n(R)$ to $M_n(R)$ when 2 and 3 are units of R.

• Journal of the Korean Mathematical Society
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• v.47 no.4
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• pp.691-704
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• 2010

Over a ring R, let $P_R$ be a finitely generated projective right R-module. Then we define the G-injector (G-projector) if $P_R$ preservers Gorenstein injective modules (Gorenstein projective modules), the Gflator if $P_R$ preservers Gorenstein flat modules. G-injector (G-flator) and G-injector are characterized focus primarily on the cases where R is a Gorenstein ring, and under this condition we also study the relations between the injector (projector, flator) and the G-injector (G-projector, G-flator). Over any ring we also give the characteristics of G-injector (G-flator) by the Gorenstein injective (Gorenstein flat) dimensions of modules.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.581-589
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• 2006

We characterize the linear operators which preserve the factor rank of integer matrices. That is, if $\mathcal{M}$ is the set of all $m{\times}n$ matrices with entries in the integers and min($m,n$) > 1, then a linear operator T on $\mathcal{M}$ preserves the factor rank of all matrices in $\mathcal{M}$ if and only if T has the form either T(X) = UXV for all $X{\in}\mathcal{M}$, or $m=n$ and T(X)=$UX^tV$ for all $X{\in}\mathcal{M}$, where U and V are suitable nonsingular integer matrices. Other characterizations of factor rank-preservers of integer matrices are also given.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.589-597
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• 2018

For $A,B{\in}M_2(\mathbb{C})$, let the Jordan product be AB + BA and G(A) the eigenvalue inclusion set, the Gershgorin set of A. Characterization is obtained for maps ${\phi}:M_2(\mathbb{C}){\rightarrow}M_2(\mathbb{C})$ satisfying $$G[{\phi}(A){\phi}(B)+{\phi}(B){\phi}(A)]=G(AB+BA)$$ for all matrices A and B. In fact, it is shown that such a map has the form ${\phi}(A)={\pm}(PD)A(PD)^{-1}$, where P is a permutation matrix and D is a unitary diagonal matrix in $M_2(\mathbb{C})$.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.353-365
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• 1994

During the past century, one of the most active and continuing subjects in matrix theory has been the study of those linear operators on matrices that leave certain properties or subsets invariant. Such questions ar usually called "Linear Preserver Problems".

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.501-507
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• 2005

A Boolean matrix with rank 1 is factored as a left factor and a right factor. The perimeter of a rank-1 Boolean matrix is defined as the number of nonzero entries in the left factor and the right factor of the given matrix. We obtain new characterizations of rank preservers, in terms of perimeter, of Boolean matrices.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.169-178
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• 2007

We consider the set of $n{\times}n$ idempotent matrices and we characterize the linear operators that preserve idempotent matrices over Boolean algebras. We also obtain characterizations of linear operators that preserve idempotent matrices over a chain semiring, the nonnegative integers and the nonnegative reals.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1043-1056
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• 2008

The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix ordered pairs which satisfy multiplicative properties with respect to spanning column rank of matrices over semirings.

• Kyungpook Mathematical Journal
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• v.50 no.4
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• pp.465-472
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• 2010

In this paper, we consider two extreme sets of zero-term rank sum of fuzzy matrix pairs: $$\cal{z}_1(\cal{F})=\{(X,Y){\in}\cal{M}_{m,n}(\cal{F})^2{\mid}z(X+Y)=min\{z(X),z(Y)\}\};$$ $$\cal{z}_2(\cal{F})=\{(X,Y){\in}\cal{M}_{m,n}(\cal{F})^2{\mid}z(X+Y)=0\}$$. We characterize the linear operators that preserve these two extreme sets of zero-term rank sum of fuzzy matrix pairs.