Essential ideals, homomorphically closed classes and their radicals

Conditions are given on a radical (semisimple) property to ensure the existence of a construction for the smallest radical (semisimple) class with the property, containing a given class of rings. This generalizes earlier results on smallest hereditary or strongly hereditary radical classes, and hereditary semisimple classes. In the last section certain classes of rings are shown to admit a construction for the largest radical (semisimple) class contained in the given class. This leads to theorems on largest radical (semisimple) classes dual to the already established smallest theorems.

If R is a ring and n is an integer weMaydefine a ring Tn (R) on the same underlying additive abelian group by using the formula a * b = nab to define a new multiplication. Tn , is a functor on the category of associative rings. If C is a class of rings then, for each n, the class Cn , is defined to consist of all rings R such that Tn (R) is in C. If C is a radical class then each class Cn , is also a radical class. We consider the properties of the radical class C which are inherited by Cn , and relationships between these classes C n as n varies.

Let A be a class of rings, and let L(A) denote the lower radical class determined by A. In [3] Yu-Lee Lee showed that L(A) may b constructed in the following manner: Let H(A) be the class of all homomorphic images of rings in A.

Some Pathology for Radicals in Non-Associative Near-Rings

A class of rings is a formation whenever it contains all homomorphic images of its members and if it is subdirect product closed. In the present paper, it is shown that the lattice of all formations of finite rings is algebraic and modular. Let R be a finite commutative ring with an identity element. It is established that there is a one-to-one correspondence between the set of all invariant fuzzy prime ideals of R and the set of all fuzzy prime ideals of each ring of the formation generated by R.

On the Schröder–Bernstein property for modules

In this paper we determine the base radical and semisimple class operators for universal classes in a finite setting using algebras introduced by Puzcylowski. The six possible orders for the semigroup they form are at most 14. A complete listing of base radical and semisimple operators for a number of different universal classes is provided.

In this paper we propose a general setting in which to study the radical theory of group graded rings. If is a radical class of associative rings we consider two associated radical classes of graded rings which are denoted by G and ref. We show that if is special (respectively, normal), then both G and ref are graded special (respectively, graded normal). Also, we discuss a graded version of the ADS theorem and the termination of the Kurosh lower graded radical construction.

Certain services may be provided in a continuous, one-dimensional, ordered range of different qualities and a customer requiring a service of quality q can only be offered a quality superior or equal to q. Only a discrete set of different qualities will be offered, and a service provider will provide the same service (of fixed quality b) to all customers requesting qualities of service inferior or equal to b. Assuming all services (of quality b) are priced identically, a monopolist will choose the qualities of service and the prices that maximize profit but, under perfect competition, a service provider will choose the (inferior) quality of service that can be priced at the lowest price. Assuming significant economies of scale, two fundamentally different regimes are possible: either a number of different classes of service are offered (DC regime), or a unique class of service offers an unbounded quality of service (UC regime). The DC regime appears in one of two sub-regimes: one, BDC, in which a finite number of classes is offered, the qualities of service offered are bounded and requests for high-quality services are not met, or UDC in which an infinite number of classes of service are offered and every request is met. The types of the demand curve and of the economies of scale, and not the pace of technological change, determine the regime and the class boundaries. The price structure in the DC regime obeys very general laws.

All rings considered will be associative. For a class M of rings let UM be the class of all rings having no non-zero homomorphic image in M. A hereditary class M of prime rings is called a “special class” [see 1, p. 191] if it has the property that when I ∈ M with I an ideal of a ring R, then R/I* ∈ Mwhere I* is the annihilator of I in R, and the corresponding radical class UM is then a “special radical”. Let S be the class of all subdirectly irreducible rings with simple heart.

Humans recognize perceived continuous information by dividing it into significant segments such as words and unit motions. We believe that such unsupervised segmentation is also an important ability that robots need to learn topics such as language and motions. Hence, in this paper, we propose a method for dividing continuous time-series data into segments in an unsupervised manner. To this end, we proposed a method based on a hidden semi-Markov model with Gaussian process (GP-HSMM). If Gaussian processes, which are nonparametric models, are used, unit motion patterns can be extracted from complicated continuous motion. However, this approach requires the number of classes of segments in the time-series data in advance. To overcome this problem, in this paper, we extend GP-HSMM to a nonparametric Bayesian model by introducing a hierarchical Dirichlet process (HDP) and propose the hierarchical Dirichlet processes-Gaussian process-hidden semi-Markov model (HDP-GP-HSMM). In the nonparametric Bayesian model, an infinite number of classes is assumed and it becomes difficult to estimate the parameters naively. Instead, the parameters of the proposed HDP-GP-HSMM are estimated by applying slice sampling. In the experiments, we use various synthetic and motion-capture data to show that our proposed model can estimate a more correct number of classes and achieve more accurate segmentation than baseline methods.

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring.The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated /I-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e.all localizations of 7"are of finite injective dimension).The second result is that if A is Cohen-Macaulay, then every finitely generated Amodule can be embedded in a finitely generated /(-module of finite projective dimension if and only if A is Gorenstein.Let A denote a commutative noetherian ring.As a running hypothesis we will assume that A is a homomorphic image of a Gorenstein ring (a commutative noetherian ring R is Gorenstein if Rv has finite injective dimension as an /?p-module, for every prime ideal p of R; see [2] for other definitions equivalent to this one).However, this class of noetherian rings is rather extensive, since it contains most of the rings occurring in algebraic geometry, and, in particular, it contains the complete local rings, by Cohen's structure theorem [5].All modules will be finitely generated (f.g.), and for a f.g.A-module M and p g Spec A, depth Mp denotes (as usual) the maximal length of a Mpregular sequence in pAp, while grade M is the maximal length of an /(-regular sequence in Ann M. Furthermore, recall that a f.g.nonzero Amodule Q is said to be a Gorenstein module if Í2 pointwise is of finite injective dimension (i.e.inj dim Op< co for all p e Supp O) and depth Í2p= depth Ap for all p e Supp O. Since A is a homomorphic image of a Gorenstein ring, by assumption, there exists a Gorenstein /1-module which is of rank one (i.e.Hom(i2, D.) = A), if A is a Cohen-Macaulay ring (cf.Sharp [II], and see [10], [6] and [7] for further properties of Gorenstein modules).Theorem I.A is a Cohen-Macaulay ring if and only if every f.g.Amodule can be embedded in a f.g.A-module which pointwise is of finite injective dimension.

A ring R for which x2 = 0 for all x e R is called a zerosquare ring. Zero-square rings are easily seen to be locally nilpotent. This leads to two problems: (1) constructing finitely generated zero-square rings with large index of nilpotence, and (2) investigating the structure of finitely generated zerosquare rings with given index of nilpotence. For the first problem we construct a class of zero-square rings, called free zero-square rings, whose index of nilpotence can be arbitrarily large. We show that every zero-square ring whose generators have (additive) orders dividing the orders of the generators of some free zero-square ring is a homomorphic image of the free ring. For the second problem, we assume Rn Φ 0 and obtain conditions on the additive group R+ of R (and thus also on the order of R). When n = 2, we completely characterize R+. When n > 3 we obtain the smallest possible number of generators of R+, and the smallest number of generators of order 2 in a minimal set of generators. We also determine the possible orders of R. Trivially every null ring (that is, R2 = 0) is a zero-square ring. From every nonnull commutative ring S we can make S x S x S into a nonnull zero square ring R by defining addition componentwise and multiplication by (x19 ylf Zj) x (x2, y2, z2) = (0, 0, x,y2 - x2y,) .

In this paper, we study WP-ring, that is a ring in which every nonzero weakly prime ideal is prime. Some results including the characterizations and the transfer of WP-ring property to homomorphic image and localization are given. Also, we study the possible transfer of the WP-ring property between A and $$A\propto E$$ and between A and $$A\bowtie ^{f}J$$. Our results provide new classes of commutative rings satisfying the above property.

Explicit solutions of the cubic Fermat equation are constructed in ring class fields [Formula: see text], with conductor [Formula: see text] prime to [Formula: see text], of any imaginary quadratic field [Formula: see text] whose discriminant satisfies [Formula: see text] (mod [Formula: see text]), in terms of the Dedekind [Formula: see text]-function. As [Formula: see text] and [Formula: see text] vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension [Formula: see text] of the [Formula: see text]-adic field [Formula: see text]. This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner’s conjecture: the cubic Fermat equation has a nontrivial solution in [Formula: see text] if [Formula: see text] (mod [Formula: see text]) and the class number [Formula: see text] is not divisible by [Formula: see text]. If [Formula: see text], congruence conditions for the trace of specific elements of [Formula: see text] are exhibited which imply the existence of a point of infinite order in [Formula: see text].