Journal of Algebraic Systems
Volume:5 Issue: 2, Winter - Spring 2018

Let $p:X\lo B$ be a locally trivial principal G-bundle and $\wt{p}:\wt{X}\lo B$ be a locally trivial principal $\wt{G}$-bundle. In this paper, by using the structure of principal bundles according to transition functions, we show that $\wt{G}$ is a covering group of $G$ if and only if $\wt{X}$ is a covering space of $X$. Then we conclude that a topological space $X$ with non-simply connected universal covering space has no connected locally trivial principal $\pi(X,x_0)$-bundle, for every $x_0\in X$.

Let m; n  2 be two positive integers, R a commutative ring with identity and M a unitary R-module. A proper submodule P of M is an (n 􀀀 1; n)-m-prime ((n 􀀀 1; n)-weakly prime) submodule if a1; : : : ; an􀀀1 2 R and x 2 M together with a1 : : : an􀀀1x 2 Pn(P : M)m􀀀1P (0 ̸= a1 : : : an􀀀1x 2 P) imply a1 : : : ai􀀀1ai : : : an􀀀1x 2 P, for some i 2 f1; : : : ; n􀀀1g or a1:::an􀀀1 2 (P : M). In this paper we study these submodules. Some useful results and examples concerning these types of submodules are given.

ýýThe investigation of equational compactness was initiated byý ýBanaschewski and Nelsoný. ýThey proved that pure injectivity isý ýequivalent to equational compactnessý. ýHere we define the soý ýcalled sequentially compact acts over semigroups and studyý ýsome of their categorical and homological propertiesý. ýSomeý ýBaer conditions for injectivity of S-acts are also presentedý.

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$.
In this paper, we consider some particular cases of these graphs that most of them are of importance in chemistry and study their total domination polynomials.

A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H \ T  H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H \ T  H SS , where H SS is an SS-embedded subgroup of G contained in H. In this paper, we study the influence of some SS-normal subgroups on the structure of a finite group G.

A frame $L$ is called {\it coz-dense} if $\Sigma_{coz(\alpha)}=\emptyset$ implies $\alpha=\mathbf 0$. Let $\mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $\mathcal RL$ based on minimal ideals of $\mathcal RL$ and zero sets in pointfree topology. We show that socle of $\mathcal RL$ is an essential ideal in $\mathcal RL$ if and only if the set of isolated points of $ \Sigma L$ is dense in $ \Sigma L$ if and only if the intersection of any family of essential ideals is essential in $\mathcal RL$. Besides, the counterpart of some results in the ring $C(X)$ is studied for the ring $\mathcal RL$. For example, an ideal $E$ of $\mathcal RL$ is an essential ideal if and only if $\bigcap Z[E]$ is a nowhere dense subset of $\Sigma L.$

Let $R$ be a ring and $M$ a right $R$-module. We call $M$, coretractable relative to $\overline{Z}(M)$ (for short, $\overline{Z}(M)$-coretractable) provided that, for every proper submodule $N$ of $M$ containing $\overline{Z}(M)$, there is a nonzero homomorphism $f:\dfrac{M}{N}\rightarrow M$. We investigate some conditions under which the two concepts coretractable and $\overline{Z}(M)$-coretractable, coincide. For a commutative semiperfect ring $R$, we show that $R$ is $\overline{Z}(R)$ coretractable if and only if $R$ is a Kasch ring. Some examples are provided to illustrate different concepts.