Generalizations of 𝑄𝐹-3 algebras

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

Locally Archimedian right-ordered groups and locally invariant valuation rings

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

A ring R is called a right WPF-ring (weak PF-ring) if R is a semiperfect right simple-injective ring with essential right socle. The class of right WPF-rings is broader than that of right PF-rings. In this article, we study and provide several characterizations of this new class of rings. We also show that if R is a left perfect, left and right f-injective ring, then R is QF if and only if So c 2(R) is a finitely generated right R-module if and only if R/Soc(R) is a finitely cogenerated left R-module. Some known results are obtained as corollaries.

Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ⊂ J or J ⊂ I. In this paper, we show that In(R) is not connected if and only if R ≅ M 2(D) or D 1 × D 2, for some division rings, D, D 1 and D 2. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.

The aim of the paper is to extend the class of generalized Weyl algebras (GWAs) to a larger class of rings (they are also called GWAs) that are determined by two ring endomorphisms rather than one as in the case of ‘old’ GWAs. A new class of rings, the diskew polynomial rings, is introduced that is closely related to GWAs (they are GWAs under a mild condition). Simplicity criteria are given for GWAs and diskew polynomial rings.

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincare-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Grobner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Grobner bases for commutative and non-commutative algebras. It was T. Stokes who eighteen years ago introduced Grobner left bases (GLB) and Grobner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Grobner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Grobner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Grobner bases, a review of the Grobner bases in PBW algebras, G-,and GR-algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.

Let K be a field. We obtain conditions for elements of the form {1 + a i, 1 + b j} to generate a free group of rank 2 in a quaternion algebra over K, including the case where the characteristic of the field K is 2, based on an article by Professors Jairo Gonçalves, Arnaldo Mandel and Mazi Shirvani. These results are then applied to obtain pairs of elements that freely generate a free group in many other classes of division rings, such as total fields of fractions of Ore domains and Malcev-Neumann series rings. With these results, we partially answer a conjecture by A. Lichtman, regarding the existence of non-abelian free groups in the multiplicative group of non-commutative division rings.

A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring then R is quasi-Frobenius. It was subse- quently conjectured by Carl Faith that every left (or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved the conjecture in the aArmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring with JORU 3 a 0: In this paper we construct a local ring R with JORU 3 a 0 and characterize when R is artinian or self- injective in terms of conditions on a bilinear mapping from a D-D-bimodule to D, where D is isomorphic to R=JORU. Our work shows that finding a counterexample to the Faith conjecture depends on the existence of a D-D-bimodule over a division ring D satisfying certain topological conditions.

In the first two sections of this chapter, we focus our attention on two special classes of rings, namely, local rings and semilocal rings. By definition, a ring R is local if R/rad R is a division ring, and R is semilocal if R/rad R is a semisimple ring. Thus, local rings include all division rings, and semilocal rings include all left or right artinian rings. The basic properties of local and semilocal rings are developed, respectively, in §19 and §20. We shall see, for instance, that local rings are connected with the problem of the uniqueness of Krull-Schmidt decompositions, and that semilocal rings are connected with the problem of “cancellation” of modules.

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.

We characterise artinian left duo rings via Morita duality, and show that there is a large class of such rings that possess a duality.

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.