• 대한수학회논문집
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• 제39권2호
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• pp.267-278
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• 2024

In this paper, we continue to explore an idea presented in [3] and introduce a new class of matrix rings called staircase matrix rings which has applications in noncommutative ring theory. We show that these rings preserve the notions of reduced, symmetric, reversible, IFP, reflexive, abelian rings, etc.

• 충청수학회지
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• 제32권4호
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• pp.401-408
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• 2019

A ring R is called right (resp., left) nilpotent-duo if N(R)a ⊆ aN(R) (resp., aN(R) ⊆ N(R)a) for every a ∈ R, where N(R) is the set of all nilpotents in R. In this article we provide other proofs of known results and other computations for known examples in relation with right nilpotent-duo property. Furthermore we show that the left nilpotent-duo property does not go up to a kind of matrix ring.

• 충청수학회지
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• 제14권2호
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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.

• 대한수학회보
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• 제51권3호
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• pp.813-822
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• 2014

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

• 대한수학회보
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• 제53권4호
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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 ${\Gamma}^n_{T_n(R)}$.

• 대한수학회지
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• 제47권6호
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• pp.1269-1282
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• 2010

We in this note consider a new concept, so called $\pi$-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of $\pi$-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of $\pi$-McCoy rings, observing the relations among $\pi$-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ($\pi-$)regular rings. It is proved that the n by n full matrix rings ($n\geq2$) over reduced rings are not $\pi$-McCoy, finding $\pi$-McCoy matrix rings over non-reduced rings. It is shown that the $\pi$-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of $\pi$-McCoy rings are also examined.

• 충청수학회지
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• 제16권2호
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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.

• 대한수학회보
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• 제55권3호
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• pp.913-920
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• 2018

Let R be a 2-primal strongly 2-nil-clean ring. We prove that every square matrix over R is the sum of a tripotent and a nilpotent matrices. The similar result for rings of bounded index is proved. We thereby provide a large class of rings over which every matrix is the sum of a tripotent and a nilpotent matrices.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.175-186
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• 2023

In this article we look at the property of a 2 by 2 full matrix ring over the ring of integers, of being weakly right IQNN. This generalisation of the property of being right IQNN arises from products of idempotents and nilpotents. We shown that it is, indeed, a proper generalization of right IQNN. We consider the property of beign weakly right IQNN in relation to several kinds of factorizations of a free algebra in two indeterminates over the ring of integers modulo 2.

• 대한수학회보
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• 제53권1호
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• pp.195-204
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• 2016

We prove, in this note, that a Zabavsky ring R is an elementary divisor ring if and only if R is a $B{\acute{e}}zout$ ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [7, Theorem 4], [9, Theorem 1.2.14], [11, Theorem 4] and [12, Theorem 7].