In Part I, we studied linear network coding over finite commutative rings and made comparisons to the wellstudied case of linear network coding over finite fields. Here, we consider the more general setting of linear network coding over finite (possibly non-commutative) rings and modules. We prove the following results regarding the linear solvability of directed acyclic networks over various f...

In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded k-th syzygy module over the polynomial ring. If in addition the module is Z n-graded we show that the conjecture holds in full generality. Furthermore, we give lower and upper bounds for the graded Betti numbers of graded ideals with a linear reso...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

in this paper we introduce the concept of dirac structures on (hermitian) modules and vectorbundles and deduce some of their properties. among other things we prove that there is a one to onecorrespondence between the set of all dirac structures on a (hermitian) module and the group of allautomorphisms of the module. this correspondence enables us to represent dirac structures on (hermitian)mod...

Let  be a commutative ring and  be a unitary  module. We define a semi prime sub module of a module and consider various properties of it. Also we define semi-radical of a sub module of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.

In this article, we formalize a matrix of Z-module and its properties. Specially, we formalize a matrix of a linear transformation of Z-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free Z-module V , determinant of its Gramian matrix is constant regardless of selection of its basis. Z-module is necessary for lattice problems...

A learning system composed of linear control modules, reinforcement learning modules and selection modules (a hybrid reinforcement learning system) is proposed for the fast learning of real-world control problems. The selection modules choose one appropriate control module dependent on the state. This hybrid learning system was applied to the control of a stilt-type biped robot. It learned the ...

When we cut a multiplicative proof-net of linear logic in two parts we get two modules with a certain border. We call pretype of a module the set of partitions over its border induced by Danos-Regnier switchings. The type of a module is then defined as the double orthogonal of its pretype. This is an optimal notion describing the behaviour of a module: two modules behave in the same way precise...

Most learning systems applied to problem solving have the goal of learning knowledge for solving problems more eeciently. However, very few systems have focused on the acquisition of planning operator descriptions. In this paper, we present a system that learns operator deenitions from the interaction with a completely unknown environment. In order to achieve better learning convergence, severa...