• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.207-220
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• 2004

Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idem-potency of a class of triangular Toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relational equations consistency is studied.

• East Asian mathematical journal
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• v.30 no.3
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• pp.259-270
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• 2014

For a generalized triangular matrix ring $$T=\[\array{R\;M\\0\;S}]$$, over rings R and S having only the idempotents 0 and 1 and over an (R, S)-bimodule M, we characterize all homomorphisms ${\alpha}$'s and all ${\alpha}$-derivations of T. Some of the homomorphisms are compositions of an inner homomorphism and an extended or a twisted homomorphism.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1483-1494
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• 2021

Let A and B be rings and U be a (B, A)-bimodule. If _BU has finite flat dimension, U_A has finite flat dimension and U ⊗_A C is a cotorsion left B-module for any cotorsion left A-module C, then the Gorenstein flat-cotorsion modules over the formal triangular matrix ring $T=\(\array{A&0\\U&B}\)$ are explicitly described. As an application, it is proven that each Gorenstein flat-cotorsion left T-module is flat-cotorsion if and only if every Gorenstein flat-cotorsion left A-module and B-module is flat-cotorsion. In addition, Gorenstein flat-cotorsion dimensions over the formal triangular matrix ring T are studied.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1017-1031
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• 2016

Let R be a commutative ring with the non-zero identity and n be a natural number. ${\Gamma}^n_R$ is a simple graph with $R^n{\setminus}\{0\}$ as the vertex set and two distinct vertices X and Y in $R^n$ are adjacent if and only if there exists an $n{\times}n$ lower triangular matrix A over R whose entries on the main diagonal are non-zero such that $AX^t=Y^t$ or $AY^t=X^t$, where, for a matrix B, $B^t$ is the matrix transpose of B. ${\Gamma}^n_R$ is a generalization of Cayley graph. Let $T_n(R)$ denote the $n{\times}n$ upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph ${\Gamma}^n_{T_n(R)}$.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1271-1286
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• 2015

An operator T on a complex Hilbert space H is called skew symmetric if T can be represented as a skew symmetric matrix relative to some orthonormal basis for H. In this paper, we study skew symmetric operators with eigenvalues. First, we provide an upper-triangular operator matrix representation for skew symmetric operators with nonzero eigenvalues. On the other hand, we give a description of certain skew symmetric triangular operators, which is based on the geometric relationship between eigenvectors.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.561-569
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• 1997

In this paper we shall characterize a finite triangular operator matrix with M-hyponormal operators on main diagonal. This shows in particualr that such an operator is subscalar operator. As a corollary, we get that every algebraic operator is subscalar.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.2
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• pp.19-26
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• 2001

All rings are assumed to be associative but do not necessarily have an identity. In this paper, we carry out a study of ring theoretic properties of formal triangular matrix rings. Some results are obtained on these rings concerning properties such as being $I_0$-ring, I-ring, exchange ring.

• Journal of the Chungcheong Mathematical Society
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• v.16 no.2
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• pp.79-88
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• 2003

In this paper, we study properties of a free normalizing extension ring of a FI-extending ring. We develop properties of formal triangular matrix rings and FI-extending rings. Several results on the quasi-extending modules are obtained.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.441-455
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• 2017

In this article we have determined the spectrum and fine spectrum of the lower triangular matrix B(r, s) on the Hahn sequence space h. We have also determined the approximate point spectrum, the defect spectrum and the compression spectrum of the operator B(r, s) on the sequence space h.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1387-1400
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• 2021

Suppose that $T=\(\array{A&0\\U&B}\)$ is a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. Let ℭ₁ and ℭ₂ be two classes of left A-modules, 𝔇₁ and 𝔇₂ be two classes of left B-modules. We prove that (ℭ₁, ℭ₂) and (𝔇₁, 𝔇₂) are admissible balanced pairs if and only if (p(ℭ₁, 𝔇₁), h(ℭ₂, 𝔇₂) is an admissible balanced pair in T-Mod. Furthermore, we describe when ($P^{C_1}_{D_1}$, $I^{C_2}_{D_2}$) is an admissible balanced pair in T-Mod. As a consequence, we characterize when T is a left virtually Gorenstein ring.