Max Ring Research Articles

ZWEI ‘ACHTUNDVIERZIGER VON ALTEM SCHROT UND KORNʼ SCHREIBEN ÜBER DEN VÖLKERFRÜHLING: LOUISE ASTON, MAX RING UND IHRE DARSTELLUNG DER MÄRZREVOLUTION

AbstractThe following article compares two novels whose plots are set during the March Revolution of 1848. Berlin und Breslau, 1847–49 is the novel debut of Max Ring, a now largely forgotten writer and doctor of Jewish origin who came from Upper Silesia and became the author of numerous stories, novels and non‐fiction books in the second half of the nineteenth century. He often uses the location of Berlin. In the novel Revolution und Contrerevolution, Louise Aston – poet, author and women's rights activist from Gröningen – deals with revolutionary issues. Both books were published in 1849, at a time when the consequences of these events for the future were not yet entirely foreseeable. Both Aston and Ring were supporters of the revolution. In this paper, the perspectives and views of their characters or narrators as well as the issues of narrative strategies and focalisation are analysed and discussed. Of importance here are not only the gender‐specific differences between Louise Aston and Max Ring, but primarily, in addition to similarities such as generational affiliation, the differences in their literary expression and the articulation of their political views.

Concerning the Nakayama property of a module

Abstract In this paper, we thoroughly study the Nakayama property and some related concepts. Also, we describe multiplication modules that, among other things, satisfy the Nakayama property. Next, we show that a ring 𝑅 is a Max ring if and only if all modules that can be generated by a finite or countable set have the weak Nakayama property. We prove that a ring 𝑅 is a perfect ring if and only if every module that can be generated by a finite or countable set has the Nakayama property. Finally, we present some categorical results on the aforementioned properties.

On rings admitting nonzero homomorphisms between non-projective modules

In a recent paper, Türkoğlu have studied the rings admitting nonzero homomorphisms between any two non-projective modules (property (T)). His work raises the following question: What are the rings with nonzero maps between non-projective right modules and singular simple modules? As a generalization of the property (T), we consider two families of rings: those rings with nonzero maps from any non-projective module to any singular simple module (property (T1)) and those rings with nonzero maps from any singular simple module to any non-projective module (property (T2)). Complete characterizations of both classes of rings are obtained and it is shown, in particular, that the rings satisfying property (T) are precisely those rings satisfying both properties (T1) and (T2). We show that R satisfies (T) if and only if R has a unique singular simple right R-module, and R is either two-sided Artinian hereditary serial or right completely coretractable. Furthermore, we prove that a commutative ring R has the property (T) if and only if there is a ring decomposition , where A is semisimple Artinian and B satisfies one of the following conditions: (1) B is a semi-Artinian max ring with unique singular simple right R-module. (2) B is an Artinian hereditary serial ring with unique singular simple right R-module.

On lattices associated to rings with respect to a preradical

We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these
 lattices, such as the $\sigma$-semiartinian rings, the $\sigma$-retractable modules, the $\sigma$-$V$-rings, the $\sigma$-max rings. We continue to study $\sigma$-torsion theories, $\sigma$-open classes, $\sigma$-stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical $\sigma$ is chosen as the identity preradical.

Coatomic refinable modules

In this article, we define the concept of (strongly) coatomic refinable modules as a proper generalization of (strongly) refinable modules. It is shown that: (1) every direct summand of a coatomic refinable module is coatomic refinable; (2) over a left max ring a module M is (strongly) coatomic refinable if and only if it is (strongly) refinable; (3) if a coatomic refinable module M is π-projective, then it is strongly coatomic refinable; (4) if coatomic direct summands lift modulo every coatomic submodule of M, then M is coatomic refinable.

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module MR. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ≃ A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(TT) ≤ess TT.

Characterizations of commutative max rings and some applications

Several new characterizations are given for a commutative max ring of which many are categorical.

Modules satisfying the weak Nakayama property

Let R be a commutative ring with identity. We will say that an R-module M satisfies the weak Nakayama property, if IM=M, where I is an ideal of R, implies that for any x∈M there exists a∈I such that (a−1)x=0. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R is a local ring, then R is a Max ring if and only if J(R), the Jacobson radical of R, is T-nilpotent if and only if every R-module satisfies the weak Nakayama property.

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ⊗ R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.

Rings Whose Modules are ⊕ -Cofinitely Supplemented

Abstract It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

MAX MODULES RELATIVE TO A TORSION THEORY

We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith.

Topics in torsion theory

The purpose of this thesis is to generalize to the torsion-theoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being -torsion and -torsionfree modules as well as -dense and -pure submodules. In the rst chapter we also introduce a new concept, that of a -compact module, which is basic enough to deserve a place among the preliminaries. The results that we obtain fall into three areas which are to a certain degree interrelated. The rst area is on -Max modules, which we introduce as a torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsiontheoretic counterparts which we utilize in our generalization. Furthermore, we de ne and characterize left -Max rings and apply the torsion-theoretic version of Shock's theorem to obtain a characterization of -short modules motivated by a recent article in which short modules were introduced. The second area deals with various avours of -injectivity, some known and some new. We introduce -M -injective and s-M -injective modules and examine their relationship with the known concepts of -injective and -quasiinjective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-M -injective modules, and give a generalization of Azumaya's Lemma. We also prove that every M -generated module has a -M -injective hull which is unique up to isomorphism and show how this is linked to the -quasi-injective hull. We then examine Σ-injectivity, generalizing well-known results by Faith, Albu and N ast asescu and Cailleau which provide necessary and su cient conditions for the Σinjective property, the Σ-s-M -injective property and for a direct sum of Σ-s-M -injective modules to be Σ-s-M -injective. In the third area we introduce a couple of new concepts with the aim of bringing to the torsion-theoretic setting the concept of a CS or extending module. The approach is twofold. The rst is via -CS modules which serve as a generalization of CS modules as well as -quasi-continuous, -quasi-injective and -injective modules, and the second is via s-CS modules which are a special case of CS modules. Our motivation is to provide a torsion-theoretic analogue of a well-known result by Okado which characterizes Noetherian modules. We have some partial results using s-CS modules and a nice torsion-theoretic analogue, albeit without the use of -CS or s-CS modules. We also examine the relationship between our relative versions of CS modules with those of other authors and obtain re nements to some of their results.

ON SOME LATTICES OF MODULE CLASSES

In this work we continue the discussion of the conatural classes introduced in [1]. We prove that the collection R-conat of all conatural classes of left modules over a ring R is a set, and it is a boolean lattice. Afterwards we study relationships between some lattices of module classes: R-conat and R-tors, R-her and R-quot. As a consequence of the developed theory we obtain characterizations of left MAX rings and artinian principal ideal rings.

Rings Whose Nonzero Modules Have Maximal Submodules

All rings are assumed to be associative and (except for nil-rings and some stipulated cases) to have nonzero identity elements. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module is said to be simple if it does not have nonzero proper submodules. A submodule N of a module M is called a maximal submodule (in M) if the factor module M/N is simple. A ring A is called a right max ring (or a right Bass ring) if every nonzero right A-module has a maximal submodule. A ring is said to be orthogonally finite if it does not contain an infinite set of nonzero orthogonal idempotents. Bass [11] proved that a ring with the minimum condition on principal left ideals is an orthogonally finite right max ring. Cozzens [25] and Koifman [61] constructed examples of orthogonally finite right max rings which do not satisfy the minimum condition on principal left ideals. This review contains some new and some old results related to max rings. We present the necessary notation and definitions. Let A be a ring. We denote the center of the ring A by C(A). A subset B of A is said to be central (in A) if B ⊆ C(A). If M and N are two modules, then Hom(M,N) denotes the Abelian group formed by all homomorphisms M → N . For a module M , we denote by End(M) the endomorphism ring Hom(M,M) of M . For a subset B of A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. An element a of A is said to be right regular (resp. left regular) in A if r(a) = 0 (resp. (a) = 0). A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is said to be right weakly regular if B2 = B for every right ideal B of A. A ring without nonzero nilpotent ideals is called a semiprime ring. A ring is called a prime ring if the product of any two of its nonzero ideals is not equal to zero. A ring A is said to be simple if each of its nonzero ideals coincides with A. A ring is called a domain if the product of any two of its nonzero elements is not equal to zero. For a ring A, an ideal P of A is called a prime (resp. semiprime and completely prime) ideal if the factor ring A/P is a prime ring (resp. a semiprime ring and a domain). A ring A is called a P.I.-ring (or a ring with polynomial identity) if A satisfies the polynomial identity f(x1, . . . , xn) = 0, where f(x1, . . . , xn) is a polynomial in noncommutative variables with coefficients in the ring of integers Z, and Z coincides with its ideal generated by the coefficients of f(x1, . . . , xn). A submodule N of a module M is said to be essential (in M) if N has nonzero intersection with any nonzero submodule of the module M . In this case, we say that M is an essential extension of the module N . A submodule N of a module M is called a superfluous submodule (in M) if N +M ′ = M for every proper submodule M ′ of M .

Primitively pure submodules and primitively divisible modules

A. A. Tuganbaev UDC 512.55All rings are assumed to be associative and to have a nonzero identity element. Let A be a ring,and let M be a rightA-module. A proper ideal P ofthe ringA is said to be right primitive if P is theannihilator ofa simple rightA-module (i.e., ifthere exists a maximal right idealB ofthe ringA suchthat P is the largest ideal of A contained in B). Consequently, every maximal ideal is a right and leftprimitive ideal. A submodule N ofthe moduleM is said to be primitively pure (resp. maximally pure)ifNP = NMP for every right primitive (resp. maximal) idealP ofthe ringA. A module M is said tobe primitively divisible (resp. maximally divisible)ifM = MP for every right primitive (resp. maximal)ideal P ofthe ringA.AringA is called a right max ring ifevery nonzero rightA-module has a maximalsubmodule. Maximally pure submodules and maximally divisible modules are studied in [3,4,6,7]. In [5],right max rings with polynomial identity are described. In [10], right quasi-invariant right max rings aredescribed.The main results ofthis paper are Theorems 1 and 2.Theorem1. Let A be either a P.I. ring or a right quasi-invariant ring. Then the following conditionsare equivalent.(1) A is a regular ring.(2) All submodules of any right A-module are primitively pure.(3) All submodules of any right A-module are maximally pure.(4) All principal right ideals of the ring A are primitively pure.(5) All principal right ideals of the ring A are maximally pure.Remark1. Let A be a simple ring. It is obvious that all submodules ofanyA-module M are maximallypure and primitively pure in M. Since the ring A is not necessarily regular, Theorem 1 does not hold forarbitrary rings.Theorem2. For a P.I. ring A,the following conditions are equivalent.(1) All prime ideals of the ring A are maximal ideals,andJ(A) is a right or left t-nilpotent ideal.(2) There do not exist nonzero primitively divisible right A-modules.(3) There do not exist nonzero primitively divisible left A-modules.(4) There do not exist nonzero maximally divisible right A-modules.(5) There do not exist nonzero maximally divisible left A-modules.(6) A is a right max ring.(7) A is a left max ring.The paper also contains some other assertions on primitively (resp. maximally) pure submodulesand primitively (resp. maximally) divisible modules.The proofs of Theorems 1 and 2 are decomposed into a series of lemmas. We present the necessarynotation and definitions. For a module M,theJacobson radical of M is denoted by J(M). For a moduleM and its subset X,wedenotebyr(X) the annihilator of X. For a module M, a submodule N ofM is said to be essential if NX = 0 for every nonzero submoduleX ofthe moduleM.AringA issaid to be regular if a ∈ aAa for each element a ∈ A.AringA is said to be right quasi-invariant if

Evaluation of coincidence plane definitions for improved volume resolution uniformity in PET

A new scheme for defining coincidence planes in positron emission tomography (PET) with the aim of improving the overall uniformity in volumetric spatial resolution has been investigated. The axial resolution in high-resolution PET systems rapidly degrades at off-center positions, in particular in tangential direction. This resolution loss is caused by the addition of coincidence planes with relatively large ring differences to form a single detection plane. Here, a different way of defining coincidence planes is evaluated to improve the overall spatial uniformity. This method only assigns coincidence combinations that fall within a specified slice thickness to contribute to a specific plane. The method produces overlapping axial planes, which allows a particular coincidence plane combination to contribute to several axial planes. Measurements of a point source at the center and at 20 cm off-center position showed a degradation in resolution from 4.6 to 12.4 mm full width at half-maximum (FWHM), tangentially using the conventional plane definition (max ring difference of eight). Using the new plane definition scheme (max ring difference of four), the resolution only degraded from 8.6 to 9.7 mm FWHM. This method should provide an improved resolution uniformity and image quality in particular for whole body PET.

Rings whose modules have maximal submodules

A ring $R$ is a right max ring if every right module $M\neq 0$ has at least one maximal submodule. It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module $E$ of $\operatorname{mod}$-$R$; also it suffices to check the submodules of the injective hull $E(V)$ of each simple module $V$ (Theorem 1). Another test is transfinite nilpotence of the radical of $E$ in the sense that $\operatorname{rad}^{\alpha}E=0$; equivalently, there is an ordinal $\alpha$ such that $\operatorname{rad}^{\alpha}(E(V))=0$ for each simple module $V$. This holds iff each $\operatorname{rad}^{\beta}(E(V))$ has a maximal submodule, or is zero (Theorem 2). If follows that $R$ is right max iff every nonzero (subdirectly irreducible) quasi-injective right $R$-module has a maximal submodule (Theorem 3.3). We characterize a right max ring $R$ via the endomorphism ring $\Lambda$ of any injective cogenerator $E$ of $\operatorname{mod}$-$R$; namely, $\Lambda/L$ has a minimal submodule for any left ideal $L=\operatorname{ann}_{\Lambda}M$ for a submodule (or subset) $M\ne 0$ of $E$ (Theorem 8.8). Then $\Lambda/L_0$ has socle $\ne 0$ for: (1) any finitely generated left ideal $L_0\ne\Lambda$; (2) each annihilator left ideal $L_0\ne \Lambda$; and (3) each proper left ideal $L_0=L+L'$, where $L=\operatorname{ann}_{\Lambda}M$ as above (e.g. as in (2)) and $L'$ finitely generated (Corollary 8.9A).

Locally perfect commutative rings are those whose modules have maximal submodules

R denotes a commutative ring. After Bass[B], a ring R is perfect in case every module has a projective cover. A ring R is a max ring provided that every nonzero i2-module has a maximal submodule. Bass characterized perfect rings as semilocal rings with T-nilpotent Jacobson radical J, and showed they are max rings. Moreover, Bass proved that R is perfect iff R satisfies the dec on principal ideals. Using Bass' theorems, the Hamsher-Koifman ([H],[K]) characterization of max R (see (3) ⇔(4) below), and the characterization of max R by the author via subdirectly irreducible quasi-injective R-modules, we obtain.

On some rings whose modules have maximal submodules

It is shown that a principal right ideal domain, having the property that every right module has a maximal submodule must be simple. Strong conditions satisfied by these rings are deduced giving evidence for the conjecture that they must be V-rings. We also generalize an example of Faith by showing that a subring of an infinite dimensional full linear ring, which contains the socle of that is never a left V-ring. Cozzens [1] and Kolfmann [31 gave striking examples of simple principal right ideal domains which are right V-rings (every simple module is injective). These examples answer two questions of Faith, whether there exists a nonsemisimple, simple, noetherian V-ring, and whether every V-ring is regular; and Bass' question, whether a over which every right module has a maximal submodule (a max ring) must be right perfect. Interest in these examples is enhanced by the fact that, being simple domains, they are highly nonregular and highly nonperfect, thus answering the last two questions, very reasonable ones given the state of knowledge at the time, very definitively in the negative. They show that in the noncommutative case (see [31 for the commutative case) the hypothesis R is a max ring does not seem to place very strong conditions on R. Now the most general proof of the fact that these rings are simple comes from a theorem of Faith [2, p. 1301, that any V-ring which is an order in a semisimple must be a product of simple rings. Our first purpose is to show that in fact if is a right P.I.D., as the examples cited are, then one needs only assume that is a right max in order to conclude that is simple. This result seems interesting because it seems to be an exception to the apparent weakness of the max hypothesis as cited in the previous paragraph. Received by the editors April 19, 1974. AMS (MOS) subject classifications (1970). Primary 16A04, 16A30; Secondary 16A42, 16A46.