Let T be a Noetherian ring and f a nonzerodivisor on T . We study concrete necessary and sufficient conditions for a module over R = T/(f) to be weakly liftable to T , in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over T we define the loci for certain properties: liftab...

The notions of pure ideals and pure submodules play an important role in the study of rings and their module categories. In this paper, we introduce right t-pure fuzzy ideals and pure fuzzy submodules of an R-module. We establish their basic properties, and use them to obtain fuzzy theoretic characterizations of right weakly regular rings. Among other results, we prove that a ring R is right we...

In this article we prove that an isometric multiple HNN extension of a group satisfying the falsification by fellow traveler property is almost convex. As a corollary, Wise’s example of a CAT(0) non-Hopfian group is almost convex.

The notion of a quasi-free Hilbert module over a function algebra A consisting of holomorphic functions on a bounded domain Ω in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra A. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map...

We prove that every quasi-Hopfian finitely presented structure A has a d- $$\Sigma _2$$ Scott sentence, and if in addition is computable Aut(A) satisfies natural condition, then sentence. This unifies several known results on sentences of structures it used to other not previously considered algebraic interest have sentences. In particular, we show right-angled Coxeter group finite rank as well...