We have started this class by studying the representation theory of the symmetric group Sn over the complex numbers. We finish by giving a brief introduction to the representation theory of Sn over a field F of positive characteristic p. We will also establish a connection between the representations of ŝlp and those of FSn. This connection was one of motivations to consider Kac-Moody algebra a...

Let be a module over commutative ring with identity. Before studying the concept of Strongly Pseudo Nearly Semi-2-Absorbing submodule, we need to mention ideal and basics that you study submodule. Also, introduce several characteristics submodule in classes multiplication modules other types modules. We also had no luck because is not ideal. it noted under conditions, which this faithful module...

Tensor product of irreducible modules highest weight over a semi-simple quantum group is completely reducible if and only natural contravariant form non-degenerate when restricted to the span singular vectors. We express this restriction through extremal projector providing computationally feasible criterion for complete reducibility tensor products.

Let $N$ be a submodule of a module $M$ and a minimal primary decomposition of $N$ is known‎. ‎A formula to compute Baer's lower nilradical of $N$ is given‎. ‎The relations between classical prime submodules and their nilradicals are investigated‎. ‎Some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

Given a commensurated subgroup $\Lambda$ of group $\Gamma$, we completely characterize when the inclusion $\Lambda\leq \Gamma$ is $C^*$-irreducible and provide new examples such inclusions. In particular, obtain that $\rm{PSL}(n,\mathbb{Z})\leq\rm{PGL}(n,\mathbb{Q})$ for any $n\in \mathbb{N}$, $C^*$-simple into its abstract commensurator $C^*$-irreducible. The main ingredient use fact action $\...

let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

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...

Let $R$ be a commutative ring and $M$ an $R$-module. A submodule $N$ of is called d-submodule $($resp., fd-submodule$)$ if $\ann_R(m)\subseteq \ann_R(m')$ $\ann_R(F)\subseteq \ann_R(m'))$ for some $m\in N$ finite subset $F\subseteq N)$ $m'\in M$ implies that N.$ Many examples, characterizations, properties these submodules are given. Moreover, we use them to characterize modules satisfying Prop...

SUMMARY We present a tool called MRSD (Metabolic Route Search and Design) to search and design routes based on the weighted compound transform diagraph. The search submodule returns routes between a source and product compound within seconds in the network of one or multiple organisms based on data from KEGG. The design submodule designs a route from an appointed compound in an interactive mode...

A new submodule clustering method via sparse and lowrank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic structures, e.g., image data kept as matrices. To implement clustering, the multi-way data, viewed as tensors, are represented by the proposed tensor sparse an...