We introduce an extension of the derivatives of rational expressions to expressions denoting formal power series over partially commuting variables. The expressions are purely noncommutative, however they denote partially commuting power series. The derivations (which are so-called φ-derivations) are shown to satisfy the commutation relations. Our main result states that for every so-called rig...

The submodule construction problem (SCP) as stated and formulated by Merlin and Bochmann [P. Merlin, G.V. Bochmann, On the construction of submodule specification and communication protocols, ACM Trans. Prog. Lang. Sys., 5(1) (1983) 1–25] is considered: given the specification of a system (module) and that of its n 2 1 submodules, determine the specification of the nth submodule that together w...

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...

B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*-algebras of compact operators can be characterized by the property that every norm-closed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*-module over them is automatically an orthogonal summand. We find out further generic properties of the category of Hilbert C*-modules over C*-alg...

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.

A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. 1. TAG-modules In the theory of torsion abelian groups, a di...

We say that L is weakly η-like if L/∼ is isomorphic to the natural ordering of rational numbers. We construct a computable presentation of any low weakly η-like linear ordering with no strongly η-like interval. Also we prove that there exists a noncomputably presentable low2 weakly η-like linear ordering with no strongly η-like interval.

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...