This research began as an effort to determine exactly which one-dimensional local rings have indecomposable finitely generated modules of arbitrarily large constant rank. The approach, which uses a new construction of indecomposable modules via the bimodule structure on certain Ext groups, turned out to be effective mainly for hypersurface singularities. The argument was eventually replaced by ...

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

We consider the problem of classifying noncommutative R-algebras of low rank over an arbitrary base ring R. We unify and generalize the many definitions of quaternion ring, and give several necessary and sufficient conditions which characterize them. Let R be a commutative, connected Noetherian ring (with 1). Let B be an algebra over R, an associative ring with 1 equipped with an embedding R →֒ ...

In this paper, we study subspace embedding problem and obtain the following results: 1. We extend the results of approximate matrix multiplication from the Frobenius norm to the spectral norm. Assume matrices A and B both have at most r stable rank and r̃ rank, respectively. Let S be a subspace embedding matrix with l rows which depends on stable rank, then with high probability, we have ‖ASSB−A...