Feebly Baer Rings and Modules

A right module M over a ring R is called feebly Baer if, whenever xa = 0 with x ∈ M and a ∈ R, there exists e² = e ∈ R such that xe = 0 and ea = a. The ring R is called feebly Baer if R_R is a feebly Baer module. These notions are motivated by the commutative analog discussed in a recent paper by Knox, Levy, McGovern, and Shapiro [6 Knox , M. L. , Levy , R. , McGovern , W. Wm. , Shapiro , J. ( 2009 ) Generalizations of complemented rings with applications to rings of functions. . J. Alg. Appl. 8 ( 1 ): 17 – 40 .[Crossref] , [Google Scholar]]. Basic properties of feebly Baer rings and modules are proved, and their connections with von Neumann regular rings are addressed.     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

Polynomial extensions of SCN rings need not be SCN

In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2 Kwak, T. K., Lee, M. J., Lee, Y. (2014). On sums of coe?cients of products of polynomials. Comm. Algebra 42(9):4033–4046.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]].     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

Unifying strongly clean power series rings

It is unknown whether a power series ring over a strongly clean ring is, itself, always strongly clean. Although a number of authors have shown that the above statement is true in certain special cases, the problem remains open, in general. In this article, we look at a class of strongly clean rings, which we call the optimally clean rings, over which power series are strongly clean. This condition is motivated by work in [10 Diesl, A. J., Dorsey, T. J., Garg, S., Khurana, D. (2012). A note on completeness and strongly clean rings, preprint. [Google Scholar]] and [11 Diesl, A. J., Dorsey, T. J., Iberkleid, W., LaFuente-Rodriguez, R., McGovern, W (2013). Strongly clean triangular matrices over abelian rings, preprint. [Google Scholar]]. We explore the properties of optimally clean rings and provide many examples, highlighting the role that this new class of rings plays in investigating the question of strongly clean power series.     

Special Properties of Rings of Skew Generalized Power Series

Let R be a ring, S a strictly ordered monoid, and ω: S → End(R) a monoid homomorphism. In [30 Marks , G. , Mazurek , R. , Ziembowski , M. ( 2010 ). A unified approach to various generalizations of Armendariz rings . Bull. Aust. Math. Soc. 81 : 361 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]], Marks, Mazurek, and Ziembowski study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. Following [30 Marks , G. , Mazurek , R. , Ziembowski , M. ( 2010 ). A unified approach to various generalizations of Armendariz rings . Bull. Aust. Math. Soc. 81 : 361 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]], we provide various classes of nonreduced (S, ω)-Armendariz rings, and determine radicals of the skew generalized power series ring R[[S ^≤, ω]], in terms of those of an (S, ω)-Armendariz ring R. We also obtain some characterizations for a skew generalized power series ring to be local, semilocal, clean, exchange, uniquely clean, 2-primal, or symmetric.     

On L p Resolvent Estimates for Elliptic Operators on Compact Manifolds

We prove uniform L^p estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3 Dos Santos Ferreira, D., Kenig, C., and Salo, M., 2014. On L^p resolvent estimates for Laplace-Beltrami operators on compact manifolds, Forum Math. 26 (2014), pp. 815–849.[Crossref], [Web of Science ®] , [Google Scholar]] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1 Bourgain, J., Shao, P., Sogge, C., and Yao, X., On L^p-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys., to appear.[Web of Science ®] , [Google Scholar]] very closely. We also show that spectral regions in our L^p resolvent estimates are optimal.     

On quotients of generalized Euclidean group rings

Let R = ?[C] be the integral group ring of a finite cyclic group C. Dennis et al. [4 Dennis, K., Magurn, B., Vaserstein, L. (1984). Generalized Euclidean group rings. J. Reine Angew. Math. 351:113–128.[Web of Science ®] , [Google Scholar]] proved that R is a generalized Euclidean ring in the sense of Cohn [3 Cohn, P. M. (1966). On the structure of the GL₂ of a ring. Inst. Hautes Études Sci. Publ. Math. 30:5–53.[Crossref] , [Google Scholar]], i.e., SL_n(R) is generated by the elementary matrices for all n. We prove that every proper quotient of R is also a generalized Euclidean ring.     

Residual smallness in commutative algebra

In Oman and Salminen [19 Oman, G., Salminen, A. Residually small commutative rings. J. Commut. Algebra (17 pages as a preprint, to appear). [Google Scholar]], the authors introduce and study residually small rings, defined as follows: an infinite commutative ring R with identity is residually smallif for every r∈R?{0}, there exists an ideal I_r of R such that r?I_r and |R∕I_r|<|R|. The purpose of this note is to extend our study. In particular, we continue our investigation of residually small rings and then generalize this notion to modules.     

Almost-Perfect Rings and Modules

Following [1 Amini , A. , Ershad , M. , Sharif , H. ( 2008 ). Rings over which flat covers of finitely generated modules are projective . Comm. Algebra 36 : 2862 – 2871 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.     

Elementary matrix reduction over J-stable rings

A commutative ring R is J-stable provided that R∕aR has stable range 1 for all a?J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3 Gillman, L., Henriksen, M. (1956). Some remarks about elementary divisor rings. Trans. Amer. Math. Soc. 82:362–365.[Crossref] , [Google Scholar], Theorem 8], [4 Larsen, M., Lewis, W., Shores, T. (1974). Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187:231–248.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 4.1], [7 McGovern, W. W. (2008). Bézout rings with almost stable range 1. J. Pure Appl. Algebra 212:340–348.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.7], [8 Moore, M. E. (1975). A strongly complement property of Dedekind domain. Czechoslovak Math. J. 25(100):282–283. [Google Scholar], Theorem], [9 Moore, M., Steger, A. (1971). Some results on completability in commutative rings. Pacific J. Math. 37:453–460.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.1], [14 Zabavsky, B. V. (1996). Generalized adequate rings. Ukrainian Math. J. 48:614–617.[Crossref] , [Google Scholar], Theorem 1] and [18 Zabavsky, B. V., Komarnyts’kyi, M. Y. (2010). Cohn-type theorem for adequacy and elementary divisor rings. J. Math. Sci. 167:107–111.[Crossref] , [Google Scholar], Theorem 7].     

A Regularized Hybrid Steepest Descent Method for Variational Inclusions

This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems (see, e.g., [9 I. Yamada , M. Yukawa , and M. Yamagishi ( 2011 ). Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings . In Fixed Point Algorithms for Inverse Problems in Science and Engineering ( H.H. Bauschke , R. Burachik , P.L. Combettes , V. Elser , D.R. Luke , and H. Wolkowicz , eds.), Springer-Verlag , New York , Chapter 17 , pp. 343 – 388 . [Google Scholar], 10 I. Yamada , Ogura , and N. Shirakawa ( 2002 ). A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems . In Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics ( Z. Nashed and O. Scherzer , eds.), American Mathematical Society , Providence , RI , Vol. 313 , pp. 269 – 305 . [Google Scholar]]).     

On SZ°-Ideals in Polynomial Rings

The theory of z-ideals and z°-ideals, especially as pertaining to the ideal theory of C(X), the ring of continuous functions on a completely regular Hausdorff space X, has been attended to during the recent years; see Gillman and Jerison [9 Gillman , L. , Jerison , M. ( 1976 ). Rings of Continuous Functions . New York : Springer Verlag . [Google Scholar]], Mason [18 Mason , G. ( 1989 ). Prime ideals and quotient rings of reduced rings . Math. Japan 34 : 941 – 956 . [Google Scholar]], and Azarpanah et al. [4 Azarpanah , F. , Karamzadeh , O. A. S. , Rezaei Aliabad , A. ( 2000 ). On ideals consisting entirely of zerodivisor . Comm. Algebra 28 ( 2 ): 1061 – 1073 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In this article we will consider the theory of z°-ideals as applied to the rings of polynomials over a commutative ring with identity. We introduce and study sz°-ideals (an ideal I of a ring is called sz°-ideal, if whenever S is a finite subset of I, then the intersection of all minimal prime ideals containing S is in I). In addition, we will pay attention to several annihilator conditions and find some new results. Finally, we use the two examples that appeared in Henriksen and Jerison [10 Henriksen , M. , Jerison , M. ( 1965 ). The space of minimal prime ideals of a commutative ring . Trans. Amer. Math. Soc. 115 : 110 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]] and Huckaba [12 Huckaba , J. A. ( 1988 ). Commutative Rings with Zero Divisors . Marcel-Dekker Inc . [Google Scholar]], to answer some natural questions that might arise in the literature.     

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5 Cui , J. , Chen , J. ( 2011 ). When is a 2 × 2 matrix ring over a commutative local ring quasipolar? Comm. Alg. 39 : 3212 – 3221 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] are improved or extended.     

Right and left solvable extensions of an associative Leibniz algebra

A sequence of nilpotent Leibniz algebras denoted by N_n,18 is introduced. Here n denotes the dimension of the algebra defined for n≥4; the first term in the sequence is ?₁₈ in the list of four-dimensional nilpotent Leibniz algebras introduced by Albeverio et al. [4 Albeverio, S., Omirov, B. A., Rakhimov, I. S. (2006). Classification of 4-dimensional nilpotent complex Leibniz algebras. Extr. Math. 21(3):197–210. [Google Scholar]]. Then all possible right and left solvable indecomposable extensions over the field ? are constructed so that N_n,18 serves as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

The Zero Divisor Graphs of Commutative Local Rings of Order p 4 and p 5

To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4 Bloomfield , N. , Wickham , C. ( 2010 ). Local rings with genus 2 zero divisor graph . Comm. Alg. 38 ( 8 ): 2965 – 2980 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] these classes include all local rings of order p ⁴ and p ⁵ for prime p.     

Prüfer conditions in the Nagata ring and the Serre’s conjecture ring

The Nagata ring R(X) and the Serre’s conjecture ring R?X? are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R?X?. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38 Glaz, S., Schwarz, R. (2011). Prüfer conditions in commutative rings. Arab. J. Sci. Eng. (Springer) 36:967–983.[Crossref], [Web of Science ®] [Google Scholar]] related to the transfer of Prüfer conditions to these two constructions.     

Classical Morita equivalence and linear topologies; applications to quasi-dualities

ABSTRACT

In this note,we answer a question of Hong et al. (2003 Hong , C. Y. , Kim , N. K. , Kwak , T. K. ( 2003 ). On skew Armendariz rings . Comm. Alg. 31 ( 1 ): 103 – 122 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by proving that if α is a monomorphism of a reduced ring R, and R is α-skew Armendariz, then R is α-rigid.     

On weakly clean rings

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2 Anderson, D. D., Camillo, V. P. (2002). Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30(7):3327–3336.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and were extensively investigated by Ahn and Anderson [1 Ahn, M.-S., Anderson, D. D. (2006). Weakly clean rings and almost clean rings. Rocky Mountain J. Math. 36:783–798.[Crossref], [Web of Science ®] , [Google Scholar]], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].