Advanced Developments in Cyclic Polymers: Synthesis,Applications, and Perspectives

Due to the topological effect, cyclic polymers demonstrate different and unique physical and biological properties in comparison with linear counterparts having the same molecular-weight range. With advanced synthetic and analytic technologies, cyclic polymers with different topologies, e.g. multicyclic polymers, have been reported and well characterized. For example, various cyclic DNA and related structures, such as cyclic duplexes, have been prepared conveniently by click chemistry. These types of DNA have increased resistance to enzymatic degradation and have high thermodynamic stability, and thus, have potential therapeutic applications. In addition, cyclic polymers have also been used to prepare organic–inorganic hybrids for applications in catalysis, e.g. catalyst supports. Due to developments in synthetic technology, highly pure cyclic polymers could now be produced in large scale. Therefore, we anticipate discovering more applications in the near future. Despite their promise, cyclic polymers are still less explored than linear polymers like polyolefins and polycarbonates, which are widely used in daily life. Some critical issues, including controlling the molecular weight and finding suitable applications, remain big challenges in the cyclic-polymer field. This review briefly summarizes the commonly used synthetic methodologies and focuses more on the attractive functional materials and their biological properties and potential applications.     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

Weak π-rings

In this paper we establish some characterizations for weak π-rings.     

On Litoff’s theorem

We give a short proof of Litoff’s theorem from the viewpoint of completely reducible modules.     

On primeness of general skew inverse Laurent series ring

Ever since the introduction, skew inverse Laurent series rings have kept growing in importance, as researchers characterized their properties (such as Noetherianness, Armendarizness, McCoyness, etc.) in terms of intrinsic properties of the base ring and studied their relations with other fields of mathematics, as for example quantum mechanics theory. The goal of our paper is to study the primeness and semiprimeness of general skew inverse Laurent series rings R((x^?1;σ,δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ.     

A canonical module characterization of Serre’s (R1)

In this short note, we give a characterization of domains satisfying Serre’s condition (R₁) in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author [9 Yanagawa, K. (2015). Dualizing complexes of seminormal a?ne semigroup rings and toric face rings. J. Algebra 425:367–391.[Crossref], [Web of Science ®] , [Google Scholar]] where the normality is described in terms of the “shape” of the canonical module.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.