Let R = Zq + uZq, where q = p and u = 0. In this paper, some structural properties of cyclic codes and quasi-cyclic (QC) codes over the ring R are considered. A QC code of length ln with index l over R is viewed both as in the conventional row circulant form and also as an R[x]/(x − 1)-submodule of GR(R, l)[x]/(x − 1), where GR(R, l) is the Galois extension ring of degree l over R. A necessary ...

Let R = Zq + uZq, where q = p s and u = 0. In this paper, some structural properties of cyclic codes and quasi-cyclic (QC) codes over the ring R are considered. A QC code of length ln with index l over R is viewed both as in the conventional row circulant form and also as an R[x]/(xn − 1)-submodule of GR(R, l)[x]/(xn − 1), where GR(R, l) is the Galois extension ring of degree l over R. A necess...

Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$-module $M$. A submodule $N$ of $M$ is called $T$-small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

Let $M_R$ be a module with $S=End(M_R)$. We call a submodule $K$ of $M_R$ annihilator-small if $K+T=M$, $T$ a submodule of $M_R$, implies that $ell_S(T)=0$, where $ell_S$ indicates the left annihilator of $T$ over $S$. The sum $A_R(M)$ of all such submodules of $M_R$ contains the Jacobson radical $Rad(M)$ and the left singular submodule $Z_S(M)$. If $M_R$ is cyclic, then $A_R(M)$ is the unique ...

Let $R$ be a commutative ring and $M$ be an $R$-module. In this paper, we investigate some properties of 2-absorbing submodules of $M$. It is shown that $N$ is a 2-absorbing submodule of $M$ if and only if whenever $IJLsubseteq N$ for some ideals $I,J$ of R and a submodule $L$ of $M$, then $ILsubseteq N$ or $JLsubseteq N$ or $IJsubseteq N:_RM$. Also, if $N$ is a 2-absorbing submodule of ...

We initiate the study of 1-torsion of finite modules over two-sided noetherian semiperfect rings. In particular, we give a criterion for determining when the 1-torsion submodule contains minimal generators of the module. We also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module. Some of the obtained results hold w...

A new and conceptually simple procedure is derived for the computation of the maximal reachability submodule of a given submodule of the state space of a linear discrete time system over a Noethenian ring R. The procedure is effective if R is effective and if kernels and intersections can be computed. The procedure is compared with a rather different procedure by Assan e.a. published recently.

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of $WAG2$-absorbing submodule. A number results concerning these classes submodules their homogeneous components are given.
 $N=\bigoplus _{h\in G}N_{h}$ submodule $h\in G.$ We say that $N_{h}$ is $h$-$WAG2$-absorbing $R_{e}$-module $M_{h}$ if $N_{h}\neq ...