On the nil radical

In 2015 Halina France-Jackson introduced the notion of a \({\sigma}\)-ring i.e. a ring R with the property that if I and J are ideals of R and for all \({i\in I}\), \({{j\in J}}\), there exist natural numbers m, n such that \({i^{m}j^{n} =0}\), then I = 0 or J = 0. It is shown that \({\sigma}\) is a special class which coincides with the class \({\rho}\) of all prime nil-semisimple rings. This implies that the upper nil radical of any ring R is the intersection of all ideals I of the ring such that R/I is a \({\sigma}\)-ring. In this paper we introduce classes of rings equivalent to the \({\sigma}\)-rings and then give characterizations of the upper nil radical in terms of these rings.

The derivation of a Lagrangian invariant so-called spirality is reviewed through the Lagrangian coordinates. The value of the spirality is fixed up to a gauge transformation. The helicity conservation follows directly from this invariant. Among all ideal flows with zero helicity in a domain of flow frozen into the fluid motion, a special class is introduced. This special topological class has the possibility to transform its spirality to be identically zero everywhere in the domain. For those simply connected domains of motion, a necessary and a sufficient condition is presented for these zero spirality flows.

The main purposes of this paper are to investigate ℤ-injective rings with the representation extension property and its dual, to give a necessary and sufficient condition for a ℤ-injective ring to be an amalgamation base in the class of all rings and to determine structure of ℤ-injective Noetherian rings which are amalgamation bases. Further, in the class of all commutative rings, it is shown that a commutative ring has the representation extension property, if, and only if, it is an amalgamation base.

A polynomial in noncommuting variables is vanishing, nil or central in a ring, R R , if its value under every substitution from R R is 0, nilpotent or a central element of R R , respectively. THEOREM. If R R has no nonvanishing multilinear nil polynomials then neither has the matrix ring R n {R_n} . THEOREM. Let R R be a ring satisfying a polynomial identity modulo its nil radical N N , and let f f be a multilinear polynomial. If f f is nil in R R then f f is vanishing in R / N R/N . Applied to the polynomial x y − y x xy - yx , this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let m m be a positive integer and let F F be a field containing no m m th roots of unity other than 1. If f f is a multilinear polynomial such that for some n > 2 f m n > 2{f^m} is central in F n {F_n} , then f f is central in F n {F_n} . This is related to the (non)existence of noncrossed products among p 2 {p^2} -dimensional central division rings.

We study a special class of lattice-ordered rings and a special radical. We prove that a special radical of an l-ring is equal to the intersection of the right l-prime l-ideals for each of which the following condition holds: the quotient l-ring by the maximal l-ideal contained in a given right l-ideal belongs to the special class. The prime radical of an l-ring is equal to the intersection of the right l-semiprime l-ideals. We introduce the notion of a completely l-prime l-ideal. We prove that N 3(R) is equal to the intersection of the completely l-prime, right l-ideals of an l-ring R, where N 3(R) is the special radical of the l-ring R defined by the class of l-rings without positive divisors of zero.

The study of special radicals was begun by Andrunakievič [1]. A class of prime rings is called special if it is hereditary and closed under prime extensions. The upper radicals determined by special classes are called special. In later works Andrunakievič and Rjabuhin [2] and [3] defined the concept of a special class of modules.

Locally Archimedian right-ordered groups and locally invariant valuation rings

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

Chapter 7 continues to develop the theory of rings and studies chain conditions for ideals of a ring. The motivation came from an interesting property of the ring of integers Z that its every ascending chain of ideals terminates. This interesting property of Z was first recognized by the German mathematician Emmy Noether (1882–1935). This property leads to the concept of Noetherian rings. On the other hand, Emil Artin (1898–1962) showed that there are some rings in which every infinite descending chain of ideals terminates. Such rings are called Artinian rings. The ring Z has an infinite descending chain of ideals. This property of Z shows that Z is not an Artinian ring. In this chapter special classes of rings are also studied and deeper results on ideal theory are proved. Some finiteness condition for ideals is imposed and Noetherian rings are introduced which are versatile. The most convenient equivalent formulation of the Noetherian requirement is that the ideals of the ring satisfy the ascending chain condition. A connection between a Noetherian domain and a factorization domain is established. Moreover, the Hilbert Basis Theorem is proved. This theorem has made a revolution and has given an extensive class of Noetherian rings. Its application to algebraic geometry is also discussed. The study in this chapter culminates in rings with a descending chain condition for ideals, which determines their ideal structure.

In this paper, we introduce a new class of rings, the QB∞-rings. We investigate necessary and sufficient conditions under which an exchange ring is a QB∞-ring. The modules over an exchange QB∞-ring are studied. Also, we prove that every regular square matrix over an exchange QB∞-ring admits a diagonal reduction by pseudo-invertible matrices.

A necessary and sufficient condition is given for a semilocal domain to have the property that every finitely generated module is a direct sum of cyclic modules. Recent work by Tom Shores and Roger Wiegand [SW] and [SWB] has shed new light on the old question: What is the class of commutative rings for which every finitely generated module is a direct sum of cyclic submodules? Following their terminology, such a ring is called an FGC-ring. In this paper, we are able to settle one of their conjectures [SWB, p. 1279], thus giving a characterization of reduced FGC-rings with Noetherian maximal ideal spectrum. We show that, for R a reduced ring with Noetherian maximal ideal spectrum, R is an FGC-ring and only R is a finite direct sum of h-local Bezout domains and each localization of R is an almost maximal valuation ring. The if part of the statement above was shown in [SW] to be an easy consequence of results due to Eben Matlis [M]. Other known results on FGC-rings are reviewed in [SW]; the reader is also referred to [SW] for definitions of the concepts above. The examples constructed in [W] show that our main result answers the FGC question for a large class of rings where it was unknown. Of course it would be nice to give a complete characterization of reduced FGC-rings. This would require settling another question in [SW]: Does every FGC-ring have Noetherian maximal ideal spectrum? They show the answer is yes the ring has fewer than 2' prime ideals [SW, Corollary 6.8]. 1. The main theorem. The result stated in the introduction will easily follow from the Theorem below: Theorem. Let R be an FGC-domain. Then every nonzero prime ideal P is contained in a unique maximal ideal. Proof. Suppose R contains two distinct maximal ideals M and N with Received by the editors May 16, 1974. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 13A15. Copyright ? 1975, Anierican Mathematical Society

The problems of output regulation and tracking with internal stability for systems defined over a class of rings is considered. For systems defined over a principal ideal domain the results for the usual finite dimensional systems generalize without appreciable change. For systems over a commutative ring with identity a sufficient condition has been obtained. The condition also becomes necessary under additional constraints on the problem data.

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.

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so-called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the [Formula: see text]-injective semimodule, and the [Formula: see text]-injective semimodules through several implications, examples and counter examples. Moreover, we show that every semimodule over an arbitrary semiring can be embedded in a [Formula: see text]-[Formula: see text]-injective semimodule.

In part I we introduce vertex rings , which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over C \mathbf {C} . We show that vertex rings are characterized by Goddard axioms . These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the rôle played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced étale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.