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2022 ◽  
Vol 10 ◽  
Author(s):  
Tom Bachmann ◽  
Paul Arne Østvær

Abstract For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.


Author(s):  
Eaman Al-Khouja, Magd Alfakhory, Hamza Hakmi Eaman Al-Khouja, Magd Alfakhory, Hamza Hakmi

The object of this paper is study the notions of weak Baer and weak Rickart rings and modules. We obtained many characterizations of weak Rickart rings and provide their properties. Relations ship between a weak Rickart (weak Baer) module and its endomorphism ring are studied. We proved that a weak Baer module with no infinite set of nonzero orthogonal idempotent elements in its endomorphism ring is precisely a Baer module. In addition, the endomorphism ring of a semi-projective weak Rickart module is semi-potent and the endomorphism ring of a semi-injective coweak Rickart module is semi-potent. Furthermore, we show that a free module is weak Baer if and only if its endomorphism ring is left weak Baer.


Author(s):  
Brendan Goldsmith ◽  
Luigi Salce

AbstractThe class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.


Author(s):  
Simion Breaz ◽  
Tomasz Brzeziński

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.


2020 ◽  
Vol 15 (1) ◽  
pp. 4-17
Author(s):  
Jean-François Biasse ◽  
Xavier Bonnetain ◽  
Benjamin Pring ◽  
André Schrottenloher ◽  
William Youmans

AbstractWe propose a heuristic algorithm to solve the underlying hard problem of the CSIDH cryptosystem (and other isogeny-based cryptosystems using elliptic curves with endomorphism ring isomorphic to an imaginary quadratic order 𝒪). Let Δ = Disc(𝒪) (in CSIDH, Δ = −4p for p the security parameter). Let 0 < α < 1/2, our algorithm requires:A classical circuit of size $2^{\tilde{O}\left(\log(|\Delta|)^{1-\alpha}\right)}.$A quantum circuit of size $2^{\tilde{O}\left(\log(|\Delta|)^{\alpha}\right)}.$Polynomial classical and quantum memory.Essentially, we propose to reduce the size of the quantum circuit below the state-of-the-art complexity $2^{\tilde{O}\left(\log(|\Delta|)^{1/2}\right)}$ at the cost of increasing the classical circuit-size required. The required classical circuit remains subexponential, which is a superpolynomial improvement over the classical state-of-the-art exponential solutions to these problems. Our method requires polynomial memory, both classical and quantum.


2020 ◽  
Vol 12 (6) ◽  
pp. 66
Author(s):  
Nikken Prima Puspita ◽  
Indah Emilia Wijayanti ◽  
Budi Surodjo

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi&acute;nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.


2020 ◽  
Vol 32 (1) ◽  
pp. 49-71
Author(s):  
Alexandre Pyvovarov
Keyword(s):  

2020 ◽  
Vol 32 (3) ◽  
pp. 739-772
Author(s):  
Joachim Gräter

AbstractLet D be a division ring of fractions of a crossed product {F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order {\leq}. For D we introduce the notion of freeness with respect to {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space {F((G))} of all formal power series in G over F with respect to {\leq}. From this we obtain that all division rings of fractions of {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space {F((G))} is a skew field.


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