scholarly journals Endomorphisms and Product Bases of the Baer-Specker Group

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
E. F. Cornelius
Keyword(s):  
Endomorphism Ring ◽  
Multiplicative Group ◽  
Matrix Ring ◽  
Infinite Matrices ◽  
Specker Group ◽  
Invertible Elements

The endomorphism ring of the group of all sequences of integers, the Baer-Specker group, is isomorphic to the ring of row finite infinite matrices over the integers. The product bases of that group are represented by the multiplicative group of invertible elements in that matrix ring. All products in the Baer-Specker group are characterized, and a lemma of László Fuchs regarding such products is revisited.

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

Endoprime modules and their direct sums

2018 ◽  
Vol 17 (08) ◽  
pp. 1850155 ◽  
Author(s):  
Gangyong Lee ◽  
S. Tariq Rizvi
Keyword(s):  
Direct Summand ◽  
Prime Ring ◽  
Endomorphism Ring ◽  
Matrix Ring ◽  
Direct Sums ◽  
Direct Sum ◽  
Finite Matrix ◽  
Reduced Ring ◽  
Baer Modules ◽  
Special Classes

The purpose of this paper is to further study the endoprime modules as one of the special classes of quasi-Baer modules. As a module theoretic analogue of a prime ring, we characterize an endoprime module via its endomorphism ring and a weak retractability condition. It is shown that any direct summand of an endoprime module is an endoprime module. A characterization is obtained when a direct sum of endoprime modules is an endoprime module. It is well known that every prime ring is semicentral reduced. We prove that a column (and row) finite matrix ring over a semicentral reduced ring is also a semicentral reduced ring. Consequently, it is shown that a column (and row) finite matrix ring over a prime ring is prime. Applications and examples illustrating our results are provided.

scholarly journals Primitive Rings of Infinite Matrices

1964 ◽  
Vol 14 (1) ◽  
pp. 47-53 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Jacobson Radical ◽  
Final Section ◽  
Matrix Ring ◽  
Primitive Ideal ◽  
Alternative Proof ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Matrix Rings ◽  
Primitive Ideals ◽  
Infinite Case

E. C. Posner (5) has shown that a ring R is primitive if and only if the corresponding matrix ring Mn(R) is primitive. From this result he is able to deduce that the primitive ideals in Mn(R) are precisely those ideals of the form Mn(P), where P is a primitive ideal in R. This affords an alternative proof that the Jacobson radical of Mn(R) is Mn(J), where J is the Jacobson radical of R. But Patterson (3, 4) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R. Finally the Jacobson radical of this ring is determined.

Une correspondance entre anneaux partiels et groupes

1997 ◽  
Vol 62 (1) ◽  
pp. 60-78
Author(s):  
Patrick Simonetta
Keyword(s):  
Division Ring ◽  
Multiplicative Group ◽  
Classical Model ◽  
Solvable Groups ◽  
Additive Group ◽  
Closed Field ◽  
Unique Extension ◽  
Model Companion ◽  
Invertible Elements ◽  
Model Completion

AbstractThis work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define AS as the structure obtained from A by restraining the multiplication to A × S, and σ(AS) as the group of functions from A to A of the form x → xa + b, where (a, b) belongs to S × A. We show that AS and σ(As) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of “reduced” domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories).We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(AS) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations.We emphasize the case where the domain is a division ring K and S is its multiplicative group K×. Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2.

On the Topologically Invertible Elements of a Topological Algebra

2007 ◽  
Vol 107 (1) ◽  
pp. 73-80
Author(s):  
Hugo Arizmendi-Peimbert ◽  
Angel Carrillo-Hoyo
Keyword(s):  
Topological Algebra ◽  
Invertible Elements

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

scholarly journals A trade-off between classical and quantum circuit size for an attack against CSIDH

2020 ◽  
Vol 15 (1) ◽  
pp. 4-17
Author(s):  
Jean-François Biasse ◽  
Xavier Bonnetain ◽  
Benjamin Pring ◽  
André Schrottenloher ◽  
William Youmans
Keyword(s):  
Endomorphism Ring ◽  
State Of The Art ◽  
Quantum Circuit ◽  
List Type ◽  
Hard Problem ◽  
Trade Off ◽  
Classical State ◽  
Security Parameter ◽  
The Cost ◽  
Quadratic Order

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.

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