Automorphisms Of Rings Research Articles

LMKCDEY Revisited: Speeding Up Blind Rotation with Signed Evaluation Keys

Recently, Lee et al. introduced a novel blind rotation technique utilizing ring automorphisms also known as LMKCDEY. Among known prominent blind rotation methods, LMKCDEY stands out because of its minimal key size and efficient runtime for arbitrary secret keys, although Chillotti et al.’s approach, commonly referred to as CGGI, offers faster runtime when using binary or ternary secrets. In this paper, we propose an enhancement to LMKCDEY’s runtime by incorporating auxiliary keys that encrypt the negated values of secret key elements. Our method not only achieves faster execution than LMKCDEY but also maintains a smaller key size compared to the ternary version of CGGI. Moreover, the proposed technique is compatible with LMKCDEY with only minimal adjustments. Experimental results with OpenFHE demonstrate that our approach can improve bootstrapping runtime by 5–28%, depending on the chosen parameters.

Automorphisms of quantum polynomial rings and Drinfeld Hecke algebras

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.

Автомофизмы нильтреугольных подколец алгебр Шевалле типа $G_2$ над областями целостности. I

Let $N\Phi(K)$ be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring $K$ with the identity associated with a root system $\Phi$. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings $N\Phi(K)$. Automorphisms of the Lie algebra $N\Phi(K)$ under restrictions $K=2K=3K$ on ring $K$ are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type $A_{n}$, a description of automorphisms of Lie rings $N\Phi(K)$ over $K$ was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring $N\Phi(K)$ are described by V.M. Levchuk (1990) for type $D_4$ over $K$, and for other types by A.V. Litavrin (2017), excluding types $G_2$ and $F_4$. In this paper we describe automorphisms of a nil-triangular Lie ring of type $G_2$ over an integrity domain $K$ under the following restrictions on $K$: either $K=2K=3K$, or $3K = 0$. To study automorphisms, the upper and lower central series described in this paper, are essentially used.

Автоморфизмы нильтреугольных подколец алгебр Шевалле типа G2 над полем характеристики 2

Let NΦ(K) be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring K with the identity associated with a root system Φ (The basis of NΦ(K) consists of all elements er ∈ Φ+ of the Chevalley basis). This paper studies the well-known problem of describing automorphisms of Lie algebras and rings NΦ(K). Automorphisms of the Lie algebra NΦ(K) under restrictions K = 2K = 3K on ring K are described by Y. Cao, D. Jiang, J. Wang (Intern. J. Algebra and Computation, 2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type An, a description of automorphisms of Lie rings NΦ(K) over K was obtained by V.M. Levchuk (Siberian Mathematical Journal, 1983). Automorphisms of the Lie ring NΦ(K) are described by V.M. Levchuk (Algebra and Logic, 1990) for type D4 over K, and for other types by A.V. Litavrin (Thesis for: Cand. Sc. (Physics and Mathematics) – 01.01.06. Siberian Federal University, 2017), excluding types G2 and F4. The author (2022) obtained a description of automorphisms of Lie rings NΦ(K) of type G2 when K is an integrity domain and K = 2K = 3K or 3K = 0. In this paper we describe automorphisms of a niltriangular Lie ring of type G2 over a field K under restriction 2K = 0. To study automorphisms, the upper and lower central series described in this work are essentially used. A new non-standard automorphism was found, called an S-automorphism.

On Automorphisms of Chain Rings

Suppose R is a finite chain ring with invaraints p,n,r,k,k′,m. Suppose G is also the subset of all φ∈ Aut(R), the automorphism group of R, such that φ(πk′)=πk′, where π is a generator of the maximal ideal of R. It was found that G is a group that is, in some sense, the set of all symmetries of {πk′}. The main purpose of this article is to describe the structure of G. The subgroup G helps us understand the structure of Aut(R) in the general case which in turn provides immediate results in classifying chain rings.

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

The Golomb topology of polynomial rings, II

We study the interplay of the Golomb topology and the algebraic structure in polynomial rings K[X] over a field K. In particular, we focus on infinite fields K of positive characteristic such that the set of irreducible polynomials of K[X] is dense in the Golomb space G(K[X]). We show that, in this case, the characteristic of K is a topological invariant, and that any self-homeomorphism of G(K[X]) is the composition of multiplication by a unit and a ring automorphism of K[X].

Automorphisms of matrix rings

Automorphisms of matrix rings

Characterizations and Properties of Monic Principal Skew Codes over Rings

Let A be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, δ a σ -derivation of A , and consider the skew-polynomial ring A X ; σ , δ . When A is a finite field, a Galois ring, or a general ring, some fairly recent literature used A X ; σ , δ to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principal f , σ , δ -codes, where f ∈ A X ; σ , δ is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code C . When A is a finite commutative ring and σ is a ring automorphism of A , we also give recursive formulas for the entries of a parity-check matrix of C . Also, in this case, with δ = 0 , we present a characterization of monic principal σ -codes whose dual codes are also monic principal σ -codes, and we deduce a characterization of self-dual monic principal σ -codes. Some corollaries concerning monic principal σ -constacyclic codes are also given, and a good number of highlighting examples is provided.

A symplectic look at the Fargues–Fontaine curve

Abstract We study a version of the Fukaya category of a symplectic 2-torus with coefficients in a locally constant sheaf of rings. The sheaf of rings includes a globally defined Novikov parameter that plays its usual role in organising polygon counts by area. It also includes a ring of constants whose variation around the the torus can be encoded by a pair of commuting ring automorphisms. When these constants are perfectoid of characteristic p, one of the holonomies is trivial and the other is the $p^{th}$ power map, it is possible in a limited way to specialise the Novikov parameter to 1. We prove that the Dehn twist ring defined there is isomorphic to the homogeneous coordinate ring of a scheme introduced by Fargues and Fontaine: their ‘curve of p-adic Hodge theory’ for the local field $\mathbf {F}_p(\!(z)\!)$ .

ON COMMUTING PAIR AND CENTRALISING PAIR OF AUTOMORPHISMS OF RINGS

ON COMMUTING PAIR AND CENTRALISING PAIR OF AUTOMORPHISMS OF RINGS

Descent of properties of rings and pairs of rings to fixed rings

Let G be a group acting via ring automorphisms on an integral domain R. A ring-theoretic property of R is said to be G-invariant, if \(R^G\) also has the property, where \(R^G=\{r\in R \ | \ \sigma (r)=r \ \text {for all} \ \sigma \in G\},\) the fixed ring of the action. In this paper we prove the following classes of rings are invariant under the operation \(R\rightarrow R^G:\) locally pqr domains, Strong G-domains, G-domains, Hilbert rings, S-strong rings and root-closed domains. Further let \(\mathscr {P}\) be a ring theoretic property and \(R\subseteq S\) be a ring extension. A pair of rings (R, S) is said to be a \(\mathscr {P}\)-pair, if T satisfies \(\mathscr {P}\) for each intermediate ring \(R\subseteq T\subseteq S.\) We also prove that the property \(\mathscr {P}\) descends from \((R,S)\rightarrow (R^G, S^G)\) in several cases. For instance, if \(\mathscr {P}=\) Going-down, Pseudo-valuation domain and “finite length of intermediate chains of domains”, we show each of these properties successfully transfer from \((R,S)\rightarrow (R^G, S^G).\)

An automorphic classification of real cubic curves

The action of ring automorphisms of ℝ[x,y] on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being selected.

The automorphisms of generalized cyclic Azumaya algebras

We define a nonassociative generalization of cyclic Azumaya algebras employing skew polynomial rings $D[t;\sigma]$, where $D$ is an Azumaya algebra of constant rank with center $C$ and $\sigma$ an automorphism of $D$, such that $\sigma|_{C}$ has finite order. The automorphisms of these algebras are canonically induced by ring automorphisms of the skew polynomial ring $D[t;\sigma]$ used in their construction. We achieve a description of their inner automorphisms. Results on the automorphisms of classical Azumaya algebras and central simple algebras of this type are obtained as special cases.

Expansivity on commutative rings

In this article we extend the notion of expansivity from topological dynamics to automorphisms of commutative rings with identity. We show that a ring admits a 0-expansive automorphism if and only if it is a finite product of local rings. Generalizing a well known result of compact metric spaces, we prove that if a ring admits a positively expansive automorphism then it admits finitely many maximal ideals. We prove its converse for principal ideal domains. We also consider the topological expansivity induced, in the spectrum of the ring with the Zariski topology, by an automorphism and some consequences are derived.

Compressed zero-divisor graphs of matrix rings over finite fields

ABSTRACT We extend the notion of the compressed zero-divisor graph to noncommutative rings in a way that still induces a product preserving functor from the category of finite unital rings to the category of directed graphs. For a finite field F, we investigate the properties of , the graph of the matrix ring over F, and give a purely graph-theoretic characterization of this graph when . For we prove that every graph automorphism of is induced by a ring automorphism of . We also show that for finite unital rings R and S, where S is semisimple and has no homomorphic image isomorphic to a field, if , then . In particular, this holds if with .

Automorphisms of prime rings that acts as derivations or anti-derivations

Automorphisms of prime rings that acts as derivations or anti-derivations

On Transformations of <i>A</i>-Hypergeometric Functions

We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.

Automorphisms of perfect power series rings

Let R be a perfect ring of characteristic p. We show that the group of continuous R-linear automorphisms of the perfect power series ring over R is generated by the automorphisms of the ordinary power series ring together with Frobenius; this answers a question of Jared Weinstein.

An identity involving automorphisms of prime rings inspired by Posner's theorem

ABSTRACTLet be a prime ring with centre , a non-zero Lie ideal of , and σ a non-trivial automorphism of such that for all . If , then it is shown that satisfies , the standard identity in four variables.