Let $G$ be a group with identity $e$. $R$ commutative $G$-graded ring non-zero identity, $S\subseteq h(R)$ multiplicatively closed subset of and $M$ graded $R$-module. In this article, we introduce study the concept $S$-1-absorbing prime submodules. A submodule $N$ $(N:_{R}M)\cap S=\emptyset$ is said to prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then eith...

In this paper, a high performance, high throughput and area efficient architecture of a multiplier for the Field Programmable Gate Array (FPGAs) is proposed. The most significant aspect of the proposed method is that, the developed multiplier architecture is based on vertical and crosswise structure of Ancient Indian Vedic Mathematics. As per The proposed architecture, for two 8-bit numbers; th...

We associate to any object in the nilpotent module category of an algebra with the 2-Calabi-Yau property a character (in the sense of [11]) and prove a multiplication formula for the characters. This formula extends a multiplication formula for the evaluation forms (in particular, dual semicanonical basis) associated to modules over a preprojective algebra given by Geiss, Leclerc and Schröer [6...

In this note we prove, for groups G fulfilling the Atiyah conjecture, a lemma relating arbitrary G-modules to free group-von-Neumann-algebra modules. As an immediate application we get properties of finitely generated pro-jective G-modules and G-modules. Other related arguments are envisaged. G; it can be defined as algebra of all (left) G-equivariant bounded linear operators on the Hilbert spa...

Let R be a commutative ring with identity. Let N and K be two submodules of a multiplication R-module M. Then N=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. In this paper we characterize some particular cases of multiplication modules by using the product of submodules.

With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split mor...