On intersections and unions of radical classes

Olson and Jenkins defined E(M) to be the class of all rings each nonzero homomorphic image of which contains either a nonzero M-ideal or an essential ideal where M, is any class of rings. E(M) was proven to be a radical class and various classes M were considered. Here the class E(M) is partitioned into two classes: H the class of all rings each nonzero homomorphic image of which has a proper essential ideal and the class H(M) of all rings each nonzero homomorphic image of which contains an M-ideal. It is shown that H is a radical class and under certain conditions H(M) is also a radical class. Various properties placed on M yield several well-known radical classes and an infinite number of supernilpotent nonspecial radical classes is constructed.

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.

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.

Some Pathology for Radicals in Non-Associative Near-Rings

Context. Nuclear rings are composed of gaseous material and/or stars and have a typical size ranging from 100 s pc to ~1 kpc. Their study is crucial to unravelling the processes involved in the evolution of the central regions of galaxies. Nuclear rings are long-lived structures, which allows the molecular gas inside to become sufficiently dense to initiate star formation. This makes them contributors to young circumnuclear populations and thus a crucial element in the study of secular evolution. However, the morphology of nuclear rings, and their potential correlations with that of the galactic hosts, remains an open subject. Aims. We examined 52 star-forming nuclear rings from the Atlas of Images of Nuclear Rings and correlated the overall galaxy morphology, in particular non-axisymmetric features, with the morphology of the nuclear ring. Methods. We divided the sample into different classes according to two visual classifications, one based on the overall morphology of the galaxy and the other based on the morphology of the nuclear ring. We defined three classes of nuclear rings: two-armed rings dominated by two dust lanes, twoarms+ rings crowded with secondary dust lanes in addition to the two main ones, and many-armed rings with multiple armlets of a similar prevalence. We employed unsharp-masked Hubble Space Telescope images to study the structure of nuclear rings. Results. We find that two-armed rings are more common in early-type grand design galaxies with strong bars. Twoarms+ rings are related to later-type and more weakly barred galaxies, both grand design and multi-armed. Lastly, many-armed rings are typically associated with later-type flocculent, multi-armed galaxies with the weakest bars. In addition, we examined the regions inside the nuclear rings and observe nuclear spirals in 28 galaxies (~90% of those galaxies for which the interior of the nuclear ring is resolved). Conclusions. We conclude that the global morphology of the host galaxy – more precisely, the presence and properties of a bar – plays a fundamental role in determining the morphology of the nuclear ring and the nuclear region. Specifically, the strength of non-axisymmetries strongly influences the morphological characteristics of a galaxy, from its innermost to its outermost regions. We suggest that two-armed rings are associated with a 180° structure forced by a strong bar, many-armed rings are associated with a very weak or absent 180° structure, and twoarms+ rings are found in intermediate cases.

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.

Abstract. In this paper, we generalize a few results of [7, 10] for lowerradical classes of rings, by using the limit ordinal construction for lowerradical classes of hemirings. 1. Introduction and preliminariesD. M. Olson and T. L. Jenksins [8] discussed general Radical Theory ofHemirings. The theory was further enriched by many authors (see [4, 12, 13]).The lower radicals were investigated by (see [5, 6, 9, 10, 11]) for radical classesof rings. Here we are interesting to generalize a several results of (see [3, 7,10]) in the frame work of hemiring which is quite different from ring theoreticalapproach discussed in (see [3, 7, 10]).A semiring ( R, + ,· ) is called a hemiring if(i) ‘+’ is commutative,(ii) there exists an element 0 ∈ R such that 0 is the identity of ( R, +) andthe zero element of ( R,· ), i.e., 0 r = r 0 = 0, ∀r ∈ R .If I is an semi-ideals of R , then we denote I ≤ R .Lower radical classes for hemirings can be constructed similar to the con-struction of lower radicals for rings (see [3, 5, 6, 9, 10,11]).First we include necessary preliminary, let

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.

In this paper, we study the class of rings in which every pure ideal is projective. We investigate the stability of this property under homomorphic image, and its transfer to various contexts of constructions such as pullbacks trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.

This paper is primarily a survey of the results on essential covers of radical classes of rings. Some problems are posed, and some new results are presented.

The Victorian era in Britain witnessed a multitude of dramatic changes. Industrialisation spurred urbanisation, leading to radical class divisions and the pauperisation of the lower class. Prudishness and moral constraints inflicted by religion were pervasive, particularly affecting marginalised groups such as sex workers, who were condemned for their actions and directed to the so-called rehabilitation institutions like Magdalene asylums. Established by the Catholic Church, these homes purported to provide refuge for fallen women but often perpetuated abuse and mistreatment (Pickard 2011, 80). Simultaneously, the era saw a surge in mass literacy and the increasing popularity of the press, which served as a medium of expression, reflecting the beliefs and views of its readership, including opinions on Magdalene asylums. This study aims to uncover these sentiments by analysing the language used to describe the institutions, their personnel, and penitents, as seen in newspapers published between 1843 and 1868. The examination of contemporary press coverage reveals a prevailing perception of the sisterhood's philanthropic efforts as life-saving, with scant criticism directed towards their work. Conversely, women in the asylums were often portrayed as burdens, their predicament viewed not as a conscious choice but as a result of fate or unintentional error. The conducted analysis demonstrates how such descriptions could reinforce societal taboos and absolve those in authority of responsibility for perpetuating economic disparities among the most vulnerable social strata.

Bézout rings with almost stable range 1