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Identities on additive mappings in semiprime rings 半素环上加性映射的恒等式
Q3 Mathematics Pub Date : 2023-01-16 DOI: 10.30970/ms.58.2.133-141
A. Z. Ansari, N. Rehman
Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : Rto R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m delta(x^n)$$ for each $x$ in $R$ and $kin {2, m, n, (n+m-1)!}$ and at last an application on Banach algebra is presented.
考虑一个环$R$,它是半素数并且具有$k$ -扭自由度。如果$F, d : Rto R$是两个相加的映射,满足$R.$中的每个$x$的代数恒等式$$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$,那么$F$将是一个广义的派生,在$R$上有一个关联的派生$d$。另一方面,本文还推导出$f$是一个广义左导,在$R$上有一个链接左导$delta$,如果它们满足$R$和$kin {2, m, n, (n+m-1)!}$中每个$x$的代数恒等式$$f(x^{n+m})=x^n f(x^m)+ x^m delta(x^n)$$,最后给出了在Banach代数上的应用。
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引用次数: 0
3D geometric moment invariants from the point of view of the classical invariant theory 从经典不变量理论的角度研究三维几何矩不变量
Q3 Mathematics Pub Date : 2023-01-16 DOI: 10.30970/ms.58.2.115-132
L. P. Bedratyuk, A. I. Bedratyuk
The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.
本文将三维几何不变量的描述问题看作经典不变量理论的问题,旨在澄清三维几何不变量与不变量理论之间的联系问题。利用群$SO(3)$和$SL(2)$是局部同构的显著事实,我们将三维几何矩不变量的推导问题简化为众所周知的经典不变量理论问题。给出了三维几何不变矩计算的精确表述,引入了同时三维几何不变矩代数的概念,并证明了它们与几种二元形式的联合$SL(2)$-不变量代数同构。为了简化不变量的计算,我们从李群的作用$SO(3)$推进到它的李代数$mathfrak{sl}_2$的作用。作者希望这些结果对图像分析和模式识别领域的研究人员有所帮助。
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引用次数: 2
On regular variation of entire Dirichlet series 关于整个Dirichlet级数的正则变分
Q3 Mathematics Pub Date : 2023-01-16 DOI: 10.30970/ms.58.2.174-181
P. Filevych, O. B. Hrybel
Consider an entire (absolutely convergent in $mathbb{C}$) Dirichlet series $F$ with the exponents $lambda_n$, i.e., of the form $F(s)=sum_{n=0}^infty a_ne^{slambda_n}$, and, for all $sigmainmathbb{R}$, put $mu(sigma,F)=max{|a_n|e^{sigmalambda_n}:nge0}$ and $M(sigma,F)=sup{|F(s)|:operatorname{Re}s=sigma}$. Previously, the first of the authors and M.M.~Sheremeta proved that if $omega(lambda)1$. In the present article we prove that the exponents of entire Dirichlet series with the same property can form an arbitrary sequence $lambda=(lambda_n)_{n=0}^infty$ not satisfying $omega(lambda)
考虑一个完整的(在$mathbb{C}$中绝对收敛的)Dirichlet级数$F$,其指数为$lambda_n$,即形式为$F(s)=sum_{n=0}^infty a_ne^{slambda_n}$,并且对于所有$sigmainmathbb{R}$,放入$mu(sigma,F)=max{|a_n|e^{sigmalambda_n}:nge0}$和$M(sigma,F)=sup{|F(s)|:operatorname{Re}s=sigma}$。此前,第一个作者和M.M. Sheremeta证明了$omega(lambda)1$。本文证明了具有相同性质的整个狄利克雷级数的指数可以形成一个任意序列$lambda=(lambda_n)_{n=0}^infty$不满足$omega(lambda)
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引用次数: 1
On a semitopological semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ when a family $mathscr{F}$ consists of inductive non-empty subsets of $omega$ 关于半拓扑半群$boldsymbol{B}_{omega}^{mathscr{F}}$当一个族$mathscr{F}$由$omega的归纳非空子集组成时$
Q3 Mathematics Pub Date : 2022-12-11 DOI: 10.30970/ms.59.1.20-28
O. Gutik, M. Mykhalenych
Let $boldsymbol{B}_{omega}^{mathscr{F}}$ be the bicyclic semigroup extension for the family $mathscr{F}$ of ${omega}$-closed subsets of $omega$ which is introduced in cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ for the family $mathscr{F}$ of inductive ${omega}$-closed subsets of $omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $boldsymbol{B}_{omega}^{mathscr{F}}$ as a proper dense subsemigroup then $Ssetminusboldsymbol{B}_{omega}^{mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $boldsymbol{B}_{omega}^{mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $boldsymbol{B}_{omega}^{mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$.
让 $boldsymbol{B}_{omega}^{mathscr{F}}$ 是族的双环半群的推广 $mathscr{F}$ 的 ${omega}$的闭子集 $omega$ 这是在 cite{Gutik-Mykhalenych=2020}我们研究半群的拓扑结构 $boldsymbol{B}_{omega}^{mathscr{F}}$ 为了家庭 $mathscr{F}$ 归纳的 ${omega}$的闭子集 $omega$。推广了双环半群拓扑化的Eberhart-Selden和Bertman-West结果 cite{Bertman-West-1976, Eberhart-Selden=1969} 并证明了半群上的每一个Hausdorff位移连续拓扑 $boldsymbol{B}_{omega}^{mathscr{F}}$ 是离散的,如果一个Hausdorff半拓扑半群 $S$ 包含 $boldsymbol{B}_{omega}^{mathscr{F}}$ 作为真密子半群 $Ssetminusboldsymbol{B}_{omega}^{mathscr{F}}$ 是一个理想的 $S$。此外,我们还证明了以下二分法:上的每一个Hausdorff局部紧移-连续拓扑 $boldsymbol{B}_{omega}^{mathscr{F}}$ 带邻接零的是紧的或离散的。由上一个结果,我们得到了上的每一个Hausdorff局部紧半群拓扑 $boldsymbol{B}_{omega}^{mathscr{F}}$ 具有伴零的半群是离散的,并且半群上的每一个Hausdorff局部紧移连续拓扑 $boldsymbol{B}_{omega}^{mathscr{F}}sqcup I$ 具有相邻紧致理想的 $I$ 是紧凑的还是理想的 $I$ 关于某些半群的局部紧拓扑的许多结果在半群的扩展上是开的 $boldsymbol{B}_{omega}^{mathscr{F}}$.
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引用次数: 0
On the trace of permuting tri-derivations on rings 环上置换三导的迹
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.20-25
D. Yılmaz, H. Yazarli
In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.Let $R$ be a $2,3$-torsion free prime ring and $F:Rtimes Rtimes Rrightarrow R$ be a permuting tri-derivation with trace $f$, $ d:Rrightarrow R$ be a derivation. In particular, the following assertions have been proved:1) if $[d(r),r]=f(r)$ for all $rin R$, then $R$ is commutative or $d=0$ (Theorem 1); 2) if $g:Rrightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $rin R$, then $F=0$ or $d=0$ (Theorem 2); 3) if $F(d(r),r,r)=f(r)$ for all $rin R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)circ f(r)=0$ for all $rin R$ (Theorem 3). In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:Rtimes Rtimes Rrightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $rin R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)circ f_{2}(r)=0$ for all $rin R$ (Theorem 4).
本文研究了素数环和半素数环上的导数、置换三导数和自同态的相互影响。设$R$是一个2,3$无扭素环,$F:R乘以R右列R$是一个有迹$F $的置换三重导数,$ d:R右列R$是一个导数。特别地,证明了下列断言:1)如果$[d(r),r]=f(r)$对于r $中的所有$r是交换的或$d=0$(定理1);2)如果$g: r右列r $是自同态使得$ f(d(r),r,r)=g(r)$对于r $中的所有$r是自同态,则$ f =0$或$d=0$(定理2);3)如果$ F (d (r), r, r) = F (r) $ r r美元,然后(i) $ F = 0美元或美元d = 0美元,美元(ii) $ $ d (r) 保监会F (r) = 0中所有$ r r美元(定理3)。在另一方面,如果存在交换tri-derivations $ F {1}, F{2}: 乘以r r rightarrow r F {1} $, $ (F {2} (r), r, r) = F {1} (r)为所有r r美元,美元$ F{1} $和$ % F {2} $ $ F{1} $的痕迹和F{2},美元,那么美元(i) $ $ F {1} = 0 F{2} = 0美元或美元,美元(ii) $ $ F {1} (r) F{2} 保监会(r) = 0中所有$ r r美元(定理4)。
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引用次数: 0
Note on boundedness of the $L$-index in the direction of the composition of slice entire functions 注意在切片整个函数的合成方向上$L$-索引的有界性
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.58-68
V. Baksa, Andriy Ivanovych Bandura, T. Salo, O. Skaskiv
We study a composition of two functions belonging to a class of slice holomorphic functions in the whole $n$-dimensional complex space. The slice holomorphy in the space means that for some fixed direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ and for every point $z^0inmathbb{C}^n$ the function is holomorphic on its restriction on the slice ${z^0+tmathbf{b}: tinmathbb{C}}.$ An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution.In particular, we found conditions providing boundedness of $L$-index in the direction $mathbf{b}$ for a function $f(underbrace{Phi(z),ldots,Phi(z)}_{mtext{ times}}),$where $f: mathbb{C}^ntomathbb{C}$ is a slice entire function, $Phi: mathbb{C}^ntomathbb{C}$ is a slice entire function,${L}: mathbb{C}^ntomathbb{R}_+$ is a continuous function.The obtained results are also new in one-dimensional case, i.e. for $n=1,$ $m=1.$ They are deduced using new approach in this area analog of logarithmic criterion.For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem.
研究了整个n维复空间中属于一类片全纯函数的两个函数的复合。空间中的片全纯意味着对于某个固定的方向$mathbf{b}在mathbb{C}^nsetminus mathbf{0}}$以及对于mathbb{C}^n$中的每一个点$z^0,函数在片${z^0+tmathbf{b}: t在mathbb{C}}上是全纯的。对于这些函数的联合连续性的一个附加假设允许构造一个具有有界指标的整个函数理论的类比。该类比适用于研究方向微分方程的片全纯解的性质,描述局部行为和值分布。特别地,我们发现了函数$f(underbrace{Phi(z),ldots,Phi(z)}_{mtext{times}}) $L$-index在$mathbf{b}$方向上具有有界性的条件,其中$f: mathbb{C}^n到mathbb{C}$是一个切片完整函数,$ Phi: mathbb{C}^n到mathbb{C}$是一个切片完整函数,${L}: mathbb{C}^n到mathbb{R}_+$是一个连续函数。所得结果在一维情况下也是新的,即对于$n=1,$ $m=1。它们是用类似对数准则的新方法推导出来的。对于复合中的一类非消失外函数,用对数判据得到的充分条件比用海曼定理得到的条件弱。
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引用次数: 1
Minimal growth of entire functions with prescribed zeros outside exceptional sets 例外集外具有指定零的整函数的最小增长
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.51-57
I. Andrusyak, P. Filevych, O. Oryshchyn
Let $h$ be a positive continuous increasing to $+infty$ function on $mathbb{R}$. It is proved that for an arbitrary complex sequence $(zeta_n)$ such that $0<|zeta_1|le|zeta_2|ledots$ and $zeta_ntoinfty$ as $ntoinfty$, there exists an entire function $f$ whose zeros are the $zeta_n$, with multiplicities taken into account, for which$$ln m_2(r,f)=o(N(r)),quad rnotin E, rto+infty.$$with a set $E$ satisfying $int_{Ecap(1,+infty)}h(r)dr<+infty$, if and only if $ln h(r)=O(ln r)$ as $rto+infty$.Here $N(r)$ is the integrated counting function of the sequence $(zeta_n)$ and$$m_2(r,f)=left(frac{1}{2pi}int_0^{2pi}|ln|f(re^{itheta})||^2dthetaright)^{1/2}.$$
设$h$是$mathbb{R}$上的$+infty$函数的正连续递增。证明了对于任意复序列$(zeta_n)$,使得$0<|zeta_1|le|zeta_2|ledots$和$zeta.ntoinfty$为$ntoinfty$.存在一个完整函数$f$,其零是$zeta_n$,并考虑了乘法性,其中$$ln m_2(r,f)=o(n(r)),quad r notin E,to+infty$$其中集合$E$满足$int_{Ecap(1,+infty)}h(r)dr<+infty$,当且仅当$ln h(r)=O(ln r)$为$rto+infity$。这里$N(r)$是序列$(zeta_N)$和$$m_2(r,f)=left(frac{1}{2 pi}int_0^{2 pi}|ln|f(re^{i theta})||^2d theta right)^{1/2}的积分计数函数$$
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引用次数: 0
The sharp bound of the third Hankel determinants for inverse of starlike functions with respect to symmetric points 关于对称点的星形函数逆的第三汉克尔行列式的锐界
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.45-50
B. Ráth, D. V. Krishna, K. S. Kumar, G. K. S. Viswanadh
We study the sharp bound for the third Hankel determinant for the inverse function $f$, when it belongs to of the class of starlike functions with respect to symmetric points.Let $mathcal{S}^{ast}_{s}$ be the class of starlike functions with respect to symmetric points. In the article proves the following statements (Theorem): If $fin mathcal{S}^{ast}_{s}$ thenbegin{equation*}big|H_{3,1}(f^{-1})big|leq1,end{equation*}and the result is sharp for $f(z)=z/(1-z^2).$
当反函数$f$属于关于对称点的星形函数的一类时,我们研究了它的第三个Hankel行列式的尖锐界。设$mathcal{S}^{ast}_{s}$为关于对称点的一类星形函数。在本文中证明了以下命题(定理):如果$fin mathcal{S}^{ast}_{s}$则begin{equation*}big|H_{3,1}(f^{-1})big|leq1,end{equation*},结果是尖锐的 $f(z)=z/(1-z^2).$
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引用次数: 3
On the convergence of Kurchatov-type methods using recurrent functions for solving equations 用递归函数求解方程的kurchatov型方法的收敛性
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.103-112
I. Argyros, S. Shakhno, H. Yarmola
We study a local and semi-local convergence of Kurchatov's method and its two-step modification for solving nonlinear equations under the classical Lipschitz conditions for the first-order divided differences. To develop a convergence analysis we use the approach of restricted convergence regions in a combination to our technique of recurrent functions. The semi-local convergence is based on the majorizing scalar sequences. Also, the results of the numerical experiment are given.
研究了在经典Lipschitz条件下求解一阶可分差分非线性方程的Kurchatov方法及其两步修正的局部和半局部收敛性。为了发展收敛性分析,我们将限制收敛区域的方法与我们的递归函数技术相结合。半局部收敛是基于标量序列的最大化。并给出了数值实验结果。
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引用次数: 1
On graded WAG2-absorbing submodule 梯度wag2吸收子模块
Q3 Mathematics Pub Date : 2022-10-31 DOI: 10.30970/ms.58.1.13-19
K. Al-Zoubi, Mariam Al-Azaizeh
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $WAG2$-absorbing submodule. A number of results concerning of these classes of graded submodules and their homogeneous components are given. Let $N=bigoplus _{hin G}N_{h}$ be a graded submodule of $M$ and $hin G.$ We say that $N_{h}$ is a $h$-$WAG2$-absorbing submodule of the $R_{e}$-module $M_{h}$ if $N_{h}neq M_{h}$; and whenever $r_{e},s_{e}in R_{e}$ and $m_{h}in M_{h}$ with $0neq r_{e}s_{e}m_{h}in N_{h}$, then either $%r_{e}^{i}m_{h}in N_{h}$ or $s_{e}^{j}m_{h}in N_{h}$ or $%(r_{e}s_{e})^{k}in (N_{h}:_{R_{e}}M_{h})$ for some $i,$ $j,$ $k$ $inmathbb{N}.$ We say that $N$ is {a graded }$WAG2${-absorbing submodule of }$M$ if $Nneq M$; and whenever $r_{g},s_{h}in h(R)$ and $%m_{lambda }in h(M)$ with $0neq r_{g}s_{h}m_{lambda }in N$, then either $r_{g}^{i}m_{lambda }in N$ or $s_{h}^{j}m_{lambda }in N$ or $%(r_{g}s_{h})^{k}in (N:_{R}M)$ for some $i,$ $j,$ $k$ $in mathbb{N}.$ In particular, the following assertions have been proved: Let $R$ be a $G$-graded ring, $M$ a graded cyclic $R$-module with $%Gr((0:_{R}M))=0$ and $N$ a graded submodule of $M.$ If $N$ is a graded $WAG2$% {-absorbing submodule of }$M,$ thenlinebreak $Gr((N:_{R}M))$ is a graded $WAG2$% -absorbing ideal of $R$ (Theorem 4).Let $R_{1}$ and $R_{2}$ be a $G$-graded rings. Let $R=R_{1}bigoplus R_{2}$ be a $G$-graded ring and $M=M_{1}bigoplus M_{2}$ a graded $R$-module. Let $N_{1},$ $N_{2}$ be a proper graded submodule of $M_{1}$, $M_{2}$ respectively. If $N=N_{1}bigoplus N_{2}$ is a graded $WAG2$-absorbing submodule of $M,$ then $N_{1}$ and $N_{2}$ are graded weakly primary submodule of $R_{1}$-module $M_{1},$ $R_{2}$-module $M_{2},$ respectively. Moreover, If $N_{2}neq 0$ $(N_{1}neq 0),$ then $N_{1}$ is a graded weak primary submodule of $R_{1}$-module $M_{1}$ $(N_{2}$ is a graded weak primary submodule of $R_{2}$-module  $M_{2})$ (Theorem 7).
让$G$是一个具有身份$e$的群。设$R$是$G$分次交换环,$M$是$R$分次模。本文引入了分级$WAG2$吸收子模的概念。给出了这类分次模及其齐次分量的若干结果。设G}N_{h}$中的$N=bigoplus_{h是G}中$M$和$h的分等子模。我们说$N_{h}$是$R_{e}$-模$M_{h}$的$h$-$WAG2$-吸收子模,如果$N_;并且每当r_{e}$中的$r_{e}、s_{e}和m_{h}$的$m_{h}具有$0neq r时_{e}s_{e}m_{h} 在N_{h}$中,则$%r_{e}^{i}m_{h} 在N_{h}$或$s_{e}中^{j}m_{h} 在N_{h}$或$%(r_{e}s_{e} )^{k}in(N_{h}:_{R_{e}}M_{h})$对于一些$i,$$j,$$k$$inmathbb{N}.$我们说$N$是{一个分级的}$WAG2$的{-吸收子模}$M$,如果$NneqM$;并且每当h(r)$中的$r_{g},s_{h}和h(m)$中带有$0neq r的$%m_{lambda}_{g}s_{h}m_{lambda}在N$中,然后$r_{g}^{i}m_{lambda}以N$或$s_{h}表示^{j}m_{lambda}在N$或$%(r_{g}s_{h} )^{k} in(N:_{R}M)$对于一些$i、$$j、$$k$$inmathbb{N}.$特别地,以下断言已经被证明:设$R$是$G$-分次环,$M$是具有$%Gr((0:_{R}M))=0$和$N$是$M$的分级子模块如果$N$是}$M的分级$WAG2$%{-吸收子模块,$则linebreak$Gr((N:_{R}M))$是$R$的有阶$WAG2$%-吸收理想(定理4)。设$R_{1}$和$R_{2}$是一个$G$分次环。设$R=R_{1}bigoplus R_{2}$为$G$分次环,$M=M_。设$N_{1},$$N_{2}$分别是$M_{1}$,$M_{2}$的适当分等子模。如果$N=N_{1}bigoplus N_{2}$是$M的分次$WAG2$吸收子模,$则$N_{1}$和$N_{2}美元分别是$R_{1}$-模$M_{}、$R_。此外,如果$N_{2}neq0$$(N_{1}neq 0),$则$N_{1}$是$R_{1}$-模$M_{1}$$的分次弱初等子模(N_{2}$是$R_{2}$-模块$M_{2})$的分次软弱初等子模)(定理7)。
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引用次数: 0
期刊
Matematychni Studii
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