• Journal of the Korean Mathematical Society
• /
• v.55 no.5
• /
• pp.1257-1268
• /
• 2018

In this article we first investigate a sort of unit-IFP ring by which Antoine provides very useful information to ring theory in relation with the structure of coefficients of zero-dividing polynomials. Here we are concerned with the whole shape of units and nilpotent elements in such rings. Next we study the properties of unit-IFP rings through group actions of units on nonzero nilpotent elements. We prove that if R is a unit-IFP ring such that there are finite number of orbits under the left (resp., right) action of units on nonzero nilpotent elements, then R satisfies the descending chain condition for nil left (resp., right) ideals of R and the upper nilradical of R is nilpotent.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.2
• /
• pp.285-297
• /
• 2008

For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.709-737
• /
• 2020

Let R be a prime ring of characteristic different from 2. Suppose that F, G, H and T are generalized derivations of R. Let U be the Utumi quotient ring of R and C be the center of U, called the extended centroid of R and let f(x₁, …, x_n) be a non central multilinear polynomial over C. If F(f(r₁, …, r_n))G(f(r₁, …, r_n)) - f(r₁, …, r_n)T(f(r₁, …, r_n)) = H(f(r₁, …, r_n)²) for all r₁, …, r_n ∈ R, then we describe all possible forms of F, G, H and T.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.573-586
• /
• 2018

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

• Journal of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.191-207
• /
• 2015

Let $\mathcal{R}$ be a prime ring of characteristic different from 2, $\mathcal{Q}_r$ be its right Martindale quotient ring and $\mathcal{C}$ be its extended centroid. Suppose that $\mathcal{G}$ is a nonzero generalized skew derivation of $\mathcal{R}$, ${\alpha}$ is the associated automorphism of $\mathcal{G}$, f($x_1$, ${\cdots}$, $x_n$) is a non-central multilinear polynomial over $\mathcal{C}$ with n non-commuting variables and $$\mathcal{S}=\{f(r_1,{\cdots},r_n)\left|r_1,{\cdots},r_n{\in}\mathcal{R}\}$$. If $\mathcal{G}$ acts as a Jordan homomorphism on $\mathcal{S}$, then either $\mathcal{G}(x)=x$ for all $x{\in}\mathcal{R}$, or $\mathcal{G}={\alpha}$.

• Journal of applied mathematics & informatics
• /
• v.39 no.1_2
• /
• pp.73-81
• /
• 2021

Let R be a prime ring, Qr be the right Martindale quotient ring and C be the extended centroid of R. If �� be a nonzero generalized skew derivation of R and f(x1, x2, ⋯, xn) be a multilinear polynomial over C such that (��(f(x1, x2, ⋯, xn)) - f(x1, x2, ⋯, xn)) ∈ C for all x1, x2, ⋯, xn ∈ R, then either f(x1, x2, ⋯, xn) is central valued on R or R satisfies the standard identity s4(x1, x2, x3, x4).

• Korean Journal of Mathematics
• /
• v.23 no.3
• /
• pp.337-355
• /
• 2015

We in this note consider a class of rings which is related to both power-Armendariz and central Armendariz rings, in the spirit of Armendariz and Kaplansky. We introduce central power-Armendariz as a generalization of them, and study the structure of central products of coefficients of zero-dividing polynomials. We also observe various sorts of examples to illuminate the relations between central power-Armendariz and related ring properties.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.625-631
• /
• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.15 no.4
• /
• pp.609-617
• /
• 2002

A three-dimensional(3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular rings with isosceles trapezoidal and triangular cross-sections. Displacement components u/sub s/, u/sub z/, and u/sub θin the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ψ and z directions. Potential(strain) and kinetic energies of the circular ring are formulated, and upper bound values of the frequencies we obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for the circular rings with isosceles trapezoidal and equilateral triangular cross-sections having completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. The method is applicable to thin rings, as well as thick and very thick ones.

• Communications of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.233-260
• /
• 2017

Let R be a factorial domain. In this work we investigate the connections between the arithmetic of Int(R) (i.e., the ring of integer-valued polynomials over R) and its monadic submonoids (i.e., monoids of the form {$g{\in}Int(R){\mid}g{\mid}_{Int(R)}f^k$ for some $k{\in}{\mathbb{N}}_0$} for some nonzero $f{\in}Int(R)$). Since every monadic submonoid of Int(R) is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between Int(R) and its monadic submonoids. If $R={\mathbb{Z}}$ or more generally if R has sufficiently many "nice" atoms, then we prove that the infinitude of the elasticity and the tame degree of Int(R) can be explained by using the structure of monadic submonoids of Int(R).