Computing the unit group of a commutative finite $\mathbb{Z}$-algebra

Let $R$ be a ring with identity $1$, $I(R) \neq \{0\}$ be the set of all nonunit idempotents in $R$, and $M(R)$ be the set of all primitive idempotents and 0 of $R$. We say that $I(R)$ is $additive$ if for all $e, f \in I(R)$ $(e \neq f)$, $e + f \in I(R)$. In this paper, the following are shown: (1) $I(R)$ is a finite additive set if and only if $M(R) \setminus \{0\}$ is a complete set of primitive central idempotents, char($R$) = $2$ and every nonzero idempotent of $R$ can be expressed as a sum of orthogonal primitive idempotents of $R$; (2) for a regular ring $R$ such that $I(R)$ is a finite additive set, if the multiplicative group of all units of $R$ is abelian (resp. cyclic), then $R$ is a commutative ring (resp. $R$ is a finite direct product of finite fields).

Computing primitive idempotents in finite commutative rings and applications

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel–Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.

A groupoid approach to discrete inverse semigroup algebras

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector of coefficients of the polynomial as a word on a ternary alphabet $\{-1,0 ,+1\}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λn = K ⌊x1, …, xn⌋ be the Grassmann algebra over a commutative ring K with ½ ∈ K, and δ be a skew K-derivation of Λn. It is proved that δ is a unique sum δ = δ ev + δ od of an even and odd skew derivation. Explicit formulae are given for δev and δod via the elements δ (x1), …, δ (xn). It is proved that the set of all even skew derivations of Λn coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λn. In particular, Der K(Λn) is a faithful but not simple Aut K(Λn)-module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λn are found. It is proved that the set of generic normal elements of Λn that are not units forms a single Aut K(Λn)-orbit (namely, Aut K(Λn)x1) if n is even and two orbits (namely, Aut K(Λn)x1 and Aut K(Λn)(x1 + x2 ⋯ xn)) if n is odd.

Applications of Frobenius Algebras to Representation Theory of Schur Algebras

2D perceptual grouping is a well studied area which still has its merits even in the age of powerful object recognizer, namely when no prior object knowledge is available. Often perceptual grouping mechanisms struggle with the runtime complexity stemming from the combinatorial explosion when creating larger assemblies of features, and simple thresholding for pruning hypotheses leads to cumbersome tuning of parameters. In this work we propose an incremental approach instead, which leads to an anytime method, where the system produces more results with longer runtime. Moreover the proposed approach lends itself easily to incorporation of attentional mechanisms. We show how basic 3D object shapes can thus be detected using a table plane assumption.

A matrix is a basic mathematical object that is widely used in various computations. When outsourcing expensive computations to untrusted parties, the involved matrix must be disguised before it's sent out in order to protect the privacy information in it. Some research works on secure computation had presented schemes for protecting the privacy in matrices. However, none of these schemes is defined deliberately for disguising a matrix and thus is neither highly efficient nor flexible. We propose a matrix-disguising method named FMD fast matrix disguising that has high time and space efficiency and can tune the trade-off between disguising speed and protecting strength with a parameter. FMD disguises a matrix by multiplying it with a semi-random non-singular matrix which is compose of many bar-shaped sub-matrices. Each of these sub-matrices contains a row/column of random elements with almost the same values. This special matrix structure allows FMD to disguise the original matrix with time complexity proportional to the size of the original matrix. While by adjusting the bar size of the sub-matrices, FMD can smoothly tune between high-disguising speed and high-privacy protection strength. The mathematical analysis and experimental results show that FMD is more efficient than the existing schemes and is especially suitable for resource-limited clients in privacy-preserving computation outsourcing scenarios. Copyright © 2015John Wiley & Sons, Ltd.

Injective modules for group rings and Gorenstein orders

The complexity of equivalence for commutative rings

In this paper, we study some variations of perspectivity of modules. We investigate the relationships between variations of perspectivity of a module and almost uniqueness of direct complements of a module. We show that there exist s-perspective modules, and so almost dual perspective modules, whose direct complements are not almost unique. Also we construct almost dual perspective and D3-modules, namely s-perspective modules whose direct complements are not almost unique. Moreover, we examine the direct sums of s-perspective modules and the modules whose direct complements are almost unique by using pairwise orthogonal primitive idempotents and m-local components of any module M over a commutative ring. Finally, we prove some structural results over commutative Dedekind domains. Let R be a commutative Dedekind domain with quotient field Q. Let M be a non-zero injective R-module. Then direct complements of M are almost unique if and only if either M is a torsion module such that every non-zero P-primary component is isomorphic to R(P∞) or M≅Q. As a consequence we obtain that direct complements of M are almost unique if and only if for every non-zero prime ideal P, there exist a∈{0,1} and a non-negative integer n such that TP(M)≅(R(P∞))a⊕(R/PnR) if and only if M is a D3-module, where R is a commutative Dedekind domain and M is a non-zero torsion R-module. Let R be a discrete valuation ring with maximal ideal m=pR. Let M be a non-zero reduced R-module which is not torsion-free. We show that direct complements of M are almost unique if and only if M≅R/mn for some positive integer n.

Recently the relations between tiltingtheory and trivialextension algebras are deeply studied. Let A and B be basic connected artin algebras over a commutative artin ring C. In [6] Tachikawa and Wakamatsu showed that the existence of stably equivalence between categories over the trivialextension algebras T(A)=A kDA and T(B)=Bt<DB under the assumption that there is a tiltingmodule TA with B=End(TA). In case C is a field,Hughes and Waschblisch proved that if T(B) is representation-finiteof Cartan class A, then there exists a tiltedalgebra A of Dynkin type A such that T(B)^T(A) [4]. Assem, Happel and Roldan showed that, for an algebra B over an algebraically closed field, T(B) is representation-finiteiff B is an iterated tiltedalgebra of Dynkin type [1]. However in case T(B) is not of finiterepresentation type the condition T{B) = T{A) with A hereditary does not forces B to be an iterated tiltedalgebra. Let's consider the covering A of T(A) [4]. The author proved that the condition A^B implies T(A) = T(B) and that the converse holds if T{A) is representation-finite[5]. In this paper, we prove that the condition B = A with A hereditary implies that B is an iterated algebra obtained from A. It is to be noted that in case A is not necessary representation-finite. Moreover, the proof of our theorem shows that such an algebra B is obtained by at most 3m times processes tilting from A, where m is the number of non-isomorphic primitive idempotents of A.

Let R be a commutative local finite ring. In this paper, we construct the complete set of pairwise orthogonal primitive idempotents of R[X]/ &lt;g&gt; where g is a regular polynomial in R[X]. We use this set to decompose the ring R[X]/ &lt;g&gt; and to give the structure of constacyclic codes over finite chain rings. This allows us to describe generators of the dual code C' of a constacyclic code C and to characterize non-trivial self-dual constacyclic codes over finite chain rings.

Let $\mathcal {J}$ be a nondegenerate Jordan algebra over a commutative associative ring $\Phi$ containing $\tfrac {1}{2}$. Defining the socle $\mathcal {G}$ of $\mathcal {J}$ to be the sum of all minimal inner ideals of $\mathcal {J}$, we prove that $\mathcal {G}$ is the direct sum of simple ideals of $\mathcal {J}$. Our main result is that if $\mathcal {J}$ is prime with nonzero socle, then either (i) $\mathcal {J}$ is simple unital and satisfies DCC on principal inner ideals, (ii) $\mathcal {J}$ is isomorphic to a Jordan subalgebra $\mathcal {J}’$ of the plus algebra ${A^ + }$ of a primitive associative algebra A with nonzero socle S, and $\mathcal {J}’$ contains ${S^ + }$, or (iii) $\mathcal {J}$ is isomorphic to a Jordan subalgebra $\mathcal {J}''$ of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and $\mathcal {J}''$ contains $H \cap S$. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if $\mathcal {J}$ is simple then $\mathcal {J}$ contains a completely primitive idempotent if and only if either $\mathcal {J}$ is unital and satisfies DCC on principal inner ideals or $\mathcal {J}$ is isomorphic to the Jordan algebra of symmetric elements of a $*$-simple associative algebra A with involution $*$ containing a minimal one-sided ideal.