Let $R$ be a $G$-graded ring and M $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to set all submodules $Q$ such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we topology on $\mathcal{PS}_G(M)$ having Zariski prime $Spec_G(M)$ as subspace topology, investigate several topological properties space.

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this ...

We use a fermionic extension of the bosonic module to obtain a class of B(0, N)-graded Lie superalgebras with nontrivial central extensions. 0 Introduction B(M − 1, N)-graded Lie superalgebras were first investigated and classified up to central extension by Benkart-Elduque (see also Garcia-Neher’s work in [GN]). Those root graded Lie superalgebras are a super-analog of root graded Lie algebras...

Let K be a field, S = K[x1, . . . ,xn] be the polynomial ring in n variables with coefficient in K and M be a finitely generated Zn-graded S-module. Let u ∈M be a homogeneous element in M and Z a subset of the set of variables {x1, . . . ,xn}. We denote by uK[Z] the K-subspace of M generated by all elements uv where v is a monomial in K[Z]. If uK[Z] is a free K[Z]-module, the Zn-graded K-space ...

This paper describes a multi-material virtual prototyping (MMVP) system for modelling and digital fabrication of discrete and functionally graded multi-material objects for biomedical applications. The MMVP system consists of a DMMVP module, an FGMVP module and a virtual reality (VR) simulation module. The DMMVP module is used to model discrete multi-material (DMM) objects, while the FGMVP modu...

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading monoid $G$. For each positive integer, the notions of graded-n-coherent module and are introduced. In this paper many results generalized from $n$-coherent rings to graded-$n$-coherent rings. last section, we provide necessary sufficient conditions for trivial extension graded-valuation ri...

Recall that an algebraic module is aKG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander–Re...