This paper consists of several new results about Length property the module M. F-length any Module comes from concepts like Neotherian and Artinian modules (Neo Art) with comp- sition series. We proved that Neo -module has submodule T is F-length. implies also Property. Finally, some remarks, examples definitions have been presented in this paper.

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 construction using finite affine geometries is given to show that the maximum number of edges in a τ -critical linear hypergraph is (1−o(1))τ2. This asymptotically answers a question of Roudneff [14], Aharoni and Ziv [1]. 1 This copy was printed on November 16, 2001. Submitted to Graphs and Combinatorics

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 ...

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules.Over the past several years QTAG-modules have been the subject of intense investigation yet there is much to explore.The impetus for these efforts stems from the fact that the rings considered here are almost restriction free....

In this paper branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp2n(K) to the naturally embedded subgroup Sp2n−2(K) is determined. As a corollary, inductive systems of fundamental representations for Sp∞(K) are classified. The submodule structure...

For the purposes of [K] and [KM] it became necessary to have 7× 7 matrix generators for a Sylow-3-subgroup of the Ree groups G2(q) and its normalizer. For example in [K] we wanted to show that in a seven dimensional representation the Jordan canonical form of any element of order nine is a single Jordan block of size 7. In [KM] we develop group recognition algorithms. At some stage we need to i...