On classical n-absorbing submodules

In this paper, we introduce the notion of classical n-absorbing submodules of a module M over a commutative ring R with identity, which is a generalization of classical prime submodules. A proper submodule N of M is said to be classical n-absorbing if whenever a1a2... an+1 m in M, for a1a2... an+1 in R and m in M, then there are n of the ai's whose product with m is in N. We give some basic results concerning classical n-absorbing submodules. Then the classical n-absorbing avoidance theorem for submodules is proved. Finally, classical n-absorbing submodules in several classes of modules are studied

Automated Planning of Concrete Joint Layouts with 4D-BIM

Concrete pouring represents a major critical path activity that is often affected by design limitations, structural considerations and on-site operational constraints. As such, meticulous planning is required to ensure that both the aesthetic and structural integrity of joints between cast in-situ components is achieved. Failure to adequately plan concrete pouring could lead to structural defects, construction rework or structural instability, all having major financial implications. Given the inherent complexity of large-scale construction projects, the ‘manual planning’ of concrete pouring is a challenging task and prone to human errors. Against this backdrop, this study developed 4D Building Information Management (BIM) approach to facilitate automated concrete joint positioning solution (as a proof of concept) for design professionals and contractors. The study first developed structural model in Revit, then extracted spatial information regarding all construction joints and linked them to dynamic Microsoft (MS) Excel and Matlab spreadsheets using integration facilitated by Dynamo software. Midspan points of each beam as well as floor perimeter information were gathered via codes developed in MS Excel macros. Based on the Excel outputs, Matlab programming was used to determine best concreating starting points and directions, and daily allowed concrete volume, considering limitations due to cold joints. These information were then pushed back to Revit via Dynamo in order to develop daily concrete scheduling. The developed automated programme framework offers a cost-effective and accurate methodology to address the limitations and inefficiencies of traditional methods of designing construction joints and planning pours. This framework extends the body of knowledge by introducing innovative solutions to integrate structural design considerations, constructional procedures and operational aspects for mitigating human error, and providing a novel, yet technically sound, basis for further application of BIM in structural engineering

The prime ideals intersection graph of a ring

summary:Let $R$ be a commutative ring with unity and $U(R)$ be the set of unit elements of $R$. In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$, namely, the prime ideals intersection graph of $R$, denoted by $G_{p}(R)$. The $G_{p}(R)$ is a graph with vertex set $R^*-U(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $\mathfrak{p}$ of $R$ such that $a,b\in \mathfrak{p}$. We obtain necessary and sufficient conditions on $R$ such that $G_{p}(R)$ is disconnected. We find the diameter and girth of $G_{p}(R)$. We also determine all rings whose prime ideals intersection graph is a star, path, or cycle. At the end of this paper, we study the planarity and outerplanarity of $G_{p}(R)$