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 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.     

On H-supplemented modules over a right perfect ring

In 1971, Koehler [11 Koehler, A. (1971). Quasi-projective and quasi-injective modules. Pac. J. Math. 36(3):713–720.[Crossref], [Web of Science ®] , [Google Scholar]] proved a structure theorem for quasi-projective modules over right perfect rings by using results of Wu–Jans [22 Wu, L. E. T., Jans, J. P. (1967). On quasi-projectives. Illinois J. Math. 11:439–448. [Google Scholar]]. Later Mohamed–Singh [17 Mohamed, S. H., Singh, S. (1977). Generalizations of decomposition theorems known over perfect rings. J. Aust. Math. Soc. Ser. A 24(4):496–510.[Crossref] , [Google Scholar]] studied discrete modules over right perfect rings and gave decomposition theorems for these modules. Moreover, Oshiro [18 Oshiro, K. (1983). Semiperfect modules and quasi-semiperfect modules. Osaka J. Math. 20:337–372.[Web of Science ®] , [Google Scholar]] deeply studied (quasi-)discrete modules over general rings. In this paper, we consider that decomposition theorems for H-supplemented modules with the condition (D₂) or (D₃) over right perfect rings.     

Rings Whose Nilpotent Elements form a Levitzki Radical Ring

In this article, we first discuss the relations among JH^r = 0, JH^r?1·H = 0, and JH^r·x = 0. Then we give a counterexample to the question mentioned in the Remarks of [3 Cheng, C. C., Sakkalis, T., Wang, S. S. S. (1994). A case of the Jacobian conjecture. J. Pure and Applied Algebra 96:15–18.[Crossref], [Web of Science ®] , [Google Scholar]] and prove the equivalence among JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r)) = 0, JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r?1))·H(x^(r)) = 0, and JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r))·x^(r) = 0. Finally, we give partial answer to Conjecture 2 in [4 Connell, E., Zweibel, J. (1991). Exact and coexact matrices. J. Algebra 142:110–117.[Crossref], [Web of Science ®] , [Google Scholar]].     

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

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.     

On Strongly Gorenstein FP-Injective Modules

This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1 Bennis , D. , Mahdou , N. ( 2007 ). Strongly Gorenstein projective, injective, and flat modules . J. Pure Appl. Algebra 210 : 437 – 445 .[Crossref], [Web of Science ®] , [Google Scholar]]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules.     

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*.     

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.     

Continuous Cohomology of Permutation Groups on Profinite Modules

ABSTRACT

Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991 Ahlbrandt , G. , Ziegler , M. ( 1991 ). What's so special about (?/4?)^ω? Archive for Math. Logic 31 : 115 – 132 . [CSA] [Crossref] , [Google Scholar]), Evans (1997b Evans , D. M. ( 1997b ). Computation of first cohomology groups of finite covers . J. Algebra 193 : 214 – 238 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), Evans et al. (1997 Evans , D. M. , Ivanov , A. A. , Macpherson , H. D. ( 1997 ). Finite covers . In: Evans , D. M. , ed. Model Theory of Groups and Automorphism Groups . London Mathematical Society Lecture Notes 244 . Cambridge : Cambridge Univ Press , pp. 1 – 72 .[Crossref] , [Google Scholar]), and Hodges and Pillay (1994 Hodges , W. , Pillay , A. ( 1994 ). Cohomology of structures and some problems of Ahlbrandt and Ziegler . J. London Math. Soc. 50 ( 2 ): 1 – 16 . [CSA] [Crossref] , [Google Scholar]), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

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]].     

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7 C. Castaing , P. Raynaud de Fitte , and M. Valadier , M. ( 2004 ). Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory . Kluwer Academic Publishers , Dordrecht . [Google Scholar], 11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]] and Edmond and Thibault [11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]].     

A note on some structure theorems of modules in category 0

ABSTRACT

Using the local subgroup strategy of An and O'Brien (1997 An , J. , O'Brien , E. A. ( 1997 ). A local strategy to decide the Alperin and Dade conjectures . J. Alg. 189 : 34 – 57 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), An and O'Brien (1999 An , J. , O'Brien , E. A. ( 1999 ). The Alperin and Dade conjectures for the Fischer simple group Fi₂₃ . Internat. J. Alg. Comput. 9 : 621 – 670 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we classify the radical subgroups and chains of the Fischer simple group Fi ₂₂ and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs–Navarro conjectures.

     

Supercharacter Theory Constructions Corresponding to Schur Ring Products

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].     

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.     

Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

ABSTRACT

Let ? be a complete set of Sylow subgroups of a finite group G, that is, ? contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be ?-permutable if H permutes with every member of ?. The purpose of this article is to study the influence of ?-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998 Asaad , M. ( 1998 ). On maximal subgroups of Sylow subgroups of finite groups . Comm. Algebra 26 ( 11 ): 3647 – 3652 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Asaad and Heliel (2003 Asaad , M. , Heliel , A. A. ( 2003 ). On permutable subgroups of finite groups . Arch. Math. 80 : 113 – 118 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Asaad et al. (1991 Asaad , M. , Ramadan , M. , Shaalan , A. ( 1991 ). Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group . Arch. Math. 56 : 521 – 527 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Li et al. (2003 Li , Y. , Wang , Y. , Wei , H. ( 2003 ). The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group . Arch. Math. 81 ( 3 ): 245 – 252 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Ramadan (1992 Ramadan , M. ( 1992 ). Influence of normality on maximal subgroups of Sylow subgroups of a finite group . Acta Math. Hungar. 59 ( 1–2 ): 107 – 110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), and Srinivasan (1980 Srinivasan , S. ( 1980 ). Two sufficient conditions for supersolvability of finite groups . Israel J. Math. 35 : 210 – 214 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]).