• East Asian mathematical journal
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• 제14권2호
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• pp.259-268
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• 1998

In this paper, for a commutative ring R with an identity, considering the endomorphism ring $End_R$(M) of left R-module $_RM$ which is (quasi-)injective or (quasi-)projective, some discriminations of prime endomorphism were found as follows: each epimorphism with the irreducible(or simple) kernel on a (quasi-)injective module and each monomorphism with maximal image on a (quasi-)projective module are prime. It was shown that for a field F, any given square matrix in $Mat_{n{\times}n}$(F) with maximal image and irreducible kernel is a prime matrix, furthermore, any given matrix in $Mat_{n{\times}n}$(F) for any field F can be factored into a product of prime matrices.

• 대한수학회보
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• 제35권3호
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• pp.605-613
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• 1998

We study the permanent set of the prime Boolean matrices in the semigroup of Boolean matrices. We define a class $M_n$ of prime matrices, and find all the possible permanents of the elements in $M_n$.

• 호남수학학술지
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• 제7권1호
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• pp.129-133
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• 1985

The object of this note is to discuss the relationship between some matrix transformations that naturally occur involving prime numbers in the theory of Summability.

• 대한수학회지
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• 제54권1호
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• pp.59-68
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• 2017

Let $F=R^{(n)}$ be a free R-module of finite rank $n{\geq}2$. In this paper, we characterize the prime submodules of F with at most n generators when R is a $Pr{\ddot{u}}fer$ domain. We also introduce the notion of prime matrix and we show that when R is a valuation domain, every finitely generated prime submodule of F with at least n generators is the row space of a prime matrix.

• 대한수학회보
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• 제50권6호
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• pp.2027-2034
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• 2013

In this article, it is proved that under some conditions every bijective Jordan triple product homomorphism from generalized matrix algebras onto rings is additive. As a corollary, we obtain that every bijective Jordan triple product homomorphism from $M_n(\mathcal{A})$ ($\mathcal{A}$ is not necessarily a prime algebra) onto an arbitrary ring $\mathcal{R}^{\prime}$ is additive.

• 대한수학회논문집
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• 제19권2호
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• pp.211-217
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• 2004

In a ring $R_n(K,\;J)$ where K is a commutative ring with identity and J is an ideal of K, all prime ideals of $R_n(K,\;J)$ are of the form either $M_n(P)\;o;R_n(P,\;P\;{\cap}\;J)$. Therefore there is a one to one correspondence between prime ideals of K not containing J and prime ideals of $R_n(K,\;J)$.

• 대한수학회보
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• 제20권1호
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• pp.45-49
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• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.

• Kyungpook Mathematical Journal
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• 제61권1호
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• pp.11-21
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• 2021

We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called right FD. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring R that R is a subdirect product of subdirectly irreducible right FD rings; and that R/N∗(R) is a subdirect product of right duo domains, and R/J(R) is a subdirect product of division rings, where N∗(R) (J(R)) is the prime (Jacobson) radical of R. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring R is right FD and 0 ≠ e2 = e ∈ R then eRe is also right FD, examining that the class of right FD rings is not closed under subrings.

• 대한수학회지
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• 제58권4호
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• pp.967-976
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• 2021

We obtain the list of prime knots with arc index 12 up to 16 crossings and their minimal grid diagrams. This is a continuation of the works [5] and [8] in which Cromwell matrices were generated to obtain minimal grid diagrams of all prime knots up to arc index 11. We provide minimal grid diagrams of the prime alternating knots with arc index 12. They are the 10 crossing prime alternating knots. The full list of 19,513 prime knots of arc index 12 up to 16 crossings and their minimal grid diagrams can be found in the arXiv [6].

• 열처리공학회지
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• 제19권5호
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• pp.249-256
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• 2006

Structural studies have been performed on precipitation hardening found in $L1_2$ ordered ${\gamma}^{\prime}-Ni_3(Al,Ti)$ alloys using transmission electron microscopy. A uniform solid solution of ${\gamma}^{\prime}-L1_2$ ordered phase supersaturated with Ni can be obtained by solution annealing in a suitable temperature range. The ${\gamma}^{\prime}$ phase hardens appreciably by the fine precipitation of disordered ${\gamma}$. The shape of ${\gamma}$ precipitates formed during aging is initially spherical or round-cubic and grow into platelets as aging proceeds. High resolution electron microscopy revealed that the ${\gamma}$ precipitates are perfectly coherent with the matrix ${\gamma}^{\prime}$ as long as the ${\gamma}$-precipitates are plates. The loss of coherency initiates by the introduction of dislocations at the ${\gamma}/{\gamma}^{\prime}$ interface followed by the step formation at the dislocations. The ${\gamma}$ precipitates become globular after the loss of coherency. The strength of ${\gamma}^{\prime}-Ni_3(Al,Ti)$ increases over the temperature range of experiment by the precipitation of fine ${\gamma}$ particles. The peak temperature where a maximum strength was obtained shifted to higher temperature.