Journal of the Iranian Mathematical Society
https://jims.ims.ir/
Journal of the Iranian Mathematical Societyendaily1Sat, 01 Jul 2023 00:00:00 +0430Sat, 01 Jul 2023 00:00:00 +0430On some groups without perfect factors
https://jims.ims.ir/article_172327.html
If $G$ is a group and $H, K$ are normal subgroups of $G$, $H\leq K$, then $K/H$ is said to be a $G$-perfect factor if $[K/H, G] = K/H$. If $G$ is a nilpotent group, then every non-trivial factor of $G$ is not $G$-perfect. Conversely, if $G$ is finite and all non-trivial factors of $G$ are not $G$-perfect, then $G$ is nilpotent. We study (infinite) groups with no non-trivial $G$-perfect factors. We prove that if either $G$ is a locally generalized radical group with finite section rank, or $G$ has a normal nilpotent subgroup $A$ such that $G/A$ is a locally finite group with Chernikov Sylow $p$-subgroups for every prime $p$, and $G$ has no non-trivial $G$-perfect factors, then for every prime $p$ there exists a positive integer $s_p$ such that $\zeta_{s_p}(G)$, the $s_p$-term of the upper central series of $G$, contains the Sylow $p$-subgroups of $G$, and $G/Tor(G)$ is nilpotent. In particular, $G$ is hypercentral and the hypercentral length of $G$ is at most $\omega+k$, for some positive integer $k$.Real powers and logarithms of matrices
https://jims.ims.ir/article_172664.html
We define the logarithm function and the power function for matrices. Additionally, we investigate further properties of the logarithm and power functions. Utilizing the sign function, we propose a novel approach to representing the power function. Furthermore, we compute the power function for various types of matrices, including Hermitian, orthogonal, and symmetric matrices.A generalization of the Heisenberg group
https://jims.ims.ir/article_172322.html
In our former paper we studied spectral synthesis on the Heisenberg group. This problem is closely connected with the finite dimensional representations of the Heisenberg group on the space of continuous complex valued functions. In this paper we make an attempt to generalise the Heisenberg group over any commutative topological group. Starting with a basic commutative topological group we define a non-commutative topological group whose elements are triplets consisting of an element of the basic group, an exponential on the basic group, and a nonzero complex number which serves as a scaling factor. The group operation is a combination of the addition on the basic group, the multiplication of the exponentials and the multiplication of complex nonzero numbers. Although there is no differentiability, our generalised Heisenberg group shares some basic properties with the classical one. In particular, we describe finite dimensional representations of this group on the space of continuous functions, and we show that finite dimensional translation invariant function spaces over this group consist of exponential polynomials.Certain modules with the Noetherian dimension
https://jims.ims.ir/article_172737.html
&lrm;An $R$-Module $M$ with a small submodule $S$, &lrm;such that $\frac{M}{S}$ is Noetherian, &lrm;is called a $SN$-module&lrm;. &lrm;In this paper&lrm;, &lrm;we introduce the concept of $\alpha$-$SN$-modules&lrm;, &lrm;for any ordinal $\alpha \geq 0$ ($SN$-modules are just $0$-$SN$-modules)&lrm;. &lrm;Some of the basic results of $SN$-modules extended to $\alpha$-$SN$-modules&lrm;. &lrm;It is shown that an $fs$-module $M$&lrm;, &lrm;which is $\alpha$-$SN$&lrm;, &lrm;has Noetherian dimension $\leq \alpha$&lrm;. &lrm;In particular&lrm;, &lrm;if $M$ is quotient finite-dimensional and all of its submodules are $\alpha$-$SN$&lrm;, &lrm;then $M$ has Noetherian dimension $\leq \alpha$&lrm;. &lrm;Furthermore&lrm;, &lrm;the concepts of $qn$-submodules (a proper submodule $N$ of $M$ is called a $qn$-submodule if $\frac{M}{N}$ has Noetherian dimension) and $qn$-modules are introduced&lrm;. &lrm;It is proved that if $M$ is quotient finite-dimensional and each of its submodules has at least a $qn$-submodule&lrm;, &lrm;then $M$ has Noetherian dimension. Some other results are obtained too.Rings of quotients of the ring R(L) by coz-filters
https://jims.ims.ir/article_172873.html
&lrm;In this article&lrm;, &lrm;we first introduce the concept of $z$-sets in the ring $ \mathcal{R}(L)$ of real-valued continuous functions on a completely regular frame $L$&lrm;, &lrm;and give some properties of them&lrm;. &lrm;Let $S^{-1}_{\mathcal{F}}\mathcal{R}(L)$ denote&lrm; &lrm;the ring of fractions of the ring $ \mathcal{R}(L)$&lrm;, &lrm;where ${\mathcal{F}}$ is a ${\mathrm{coz}}$-filter on $L$ and&lrm; &lrm;$S_{\mathcal{F}}$ is a multiplicatively closed subset related to ${\mathcal{F}}$&lrm;. &lrm;We show that $S^{-1}_{\mathcal{F}}\mathcal{R}(L)$&lrm; &lrm;may be realized as the direct limits of the subrings $\mathcal{R}(A)$&lrm;, &lrm;where&lrm; &lrm;$A\in&lrm; \{&lrm;\mathfrak{o}_L\big({\mathrm{coz}}(\alpha)\big)&lrm; &lrm;\colon \alpha\in S_{\mathcal{F}}\}$&lrm;. &lrm;Also&lrm;, &lrm;we show that ${\mathrm{Q_{cl} }}\mathcal{R}(L)=\mathcal{R}(L)$, &lrm;if and only if&lrm; &lrm;$\mathcal{R}(L)$ is a special saturated ring&lrm;.The story of rings of continuous functions in Ahvaz: From $C(X)$ to $C_c(X)$
https://jims.ims.ir/article_172736.html
&lrm;The author narrates how the study and research on the rings of continuous real-valued functions on a topological space initiated in Iran&lrm;. &lrm;In particular&lrm;, &lrm;almost completely all the important work of the authors&lrm;, &lrm;in this field&lrm;, &lrm;during the last four decades&lrm;, &lrm;whose related research are carried out in Ahvaz are referred to and where necessary are commented on&lrm;, &lrm;by the author&lrm;. &lrm;Also included are some related anecdotes and the contributions of Karamzadeh to the promotion of mathematics in Iran&lrm;, &lrm;during the past half-century&lrm;.On the watching number of graphs
https://jims.ims.ir/article_174727.html
Let $G=(V&lrm;, &lrm;E)$ be a simple and undirected graph&lrm;. &lrm;A watcher $\omega_i$ of $G$ is a couple of $\omega_i=(v_i&lrm;, &lrm;Z_i),$ where $v_i \in V$ and $Z_i$ is a subset of the closed neighborhood of $v_i.$ If a vertex $v \in Z_i,$ we say that $v$ is covered by $\omega_i.$ A set $W=\{\omega_1&lrm;, &lrm;\omega_2&lrm;, &lrm;\dots&lrm;, &lrm;\omega_k\}$&lrm;, &lrm;of watchers is a watching system for $G$ if the sets $L_W(v)=\{\omega_i~:~v \in Z_i&lrm; ~,~ &lrm;1 \le i \le k\}$ are non-empty and distinct&lrm;, &lrm;for every $v \in V$&lrm;. &lrm;In this paper&lrm;, &lrm;we study the watching systems of some graphs&lrm;, &lrm;and consider the watching number of Mycielski's construction of some graphs&lrm;.Conciseness on normal subgroups and new concise words from lower central and derived words
https://jims.ims.ir/article_175513.html
Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables, thus proving a generalized version of a conjecture of Azevedo and Shumyatsky. This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers. Our approach is via the study of values of $w$ on normal subgroups, and in this setting we obtain the following result: if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e.\ $w(N_1,\ldots,N_r)$, is finite.Left 3-Engel elements in groups: A survey
https://jims.ims.ir/article_172988.html
We survey left 3-Engel elements in groups.A study on the $\pi$-dual Rickart modules
https://jims.ims.ir/article_176734.html
The right $R$-module $M$ is said to be a $\pi$-dual Rickart module&lrm;, &lrm;if for every endomorphism $f:M\to M$ with projection invariant image&lrm;, &lrm;$f(M)$&lrm;, &lrm;in $M$&lrm;, &lrm;$f(M)$ is a direct summand of $M$. We show that the class of the $\pi$-dual Rickart modules contains properly the class of all $\pi$-dual Baer modules and the dual Rickart modules&lrm;. &lrm;We also investigate the transfering between a base ring $R$ and $R[x]$ (and $R[[x]]$)&lrm;. &lrm;It is shown that, in general&lrm;, &lrm;the class of $\pi$-dual Rickart modules is neither closed under direct summands nor closed under direct sums&lrm;. &lrm;We conclude the paper by giving a connection between the classes of $\pi$-dual Baer and $\pi$-lifting modules&lrm;.A survey of recent results connected with subnormal subgroups
https://jims.ims.ir/article_179702.html
In this paper we give a brief survey of some highlights from the theory of subnormal subgroups and then reveal some recent extensions of this theory due to the current authors and others.