Special classes of l-rings

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.


 
 
 The concept of a special radical for near-rings has been treated in several nonequivalent, but related, ways in the recent literature. We use the version due to K. Kaarli to establish that various prime radicals and the nil radical are special radicals on the class A of all near-rings which satisfy an extended version of the Andrunakievich Lemma. Since A includes all d.g. near-rings—and much more—these results significantly extend results previously obtained by Kaarli and by Groenewald. We also obtain special radical results for the Jacobson type radicals 30 and 3 1 , albeit on less extensive classes. Examples are given which illustrate and delimit the theory developed. 
 
 

Let \({\sigma}\) be the class of all associative rings 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 n, m such that i n j m = 0, then I = 0 or J = 0. We will show that \({\sigma}\) is a special class and the upper radical \({\mathcal{U}(\sigma)}\) determined by \({\sigma}\) is equal to the nil radical \({\mathcal{N}}\) of Koethe. This implies that the nil radical \({\mathcal{N}(R)}\) of any ring R is the intersection of all \({\sigma}\)-ideals I of R, that is, ideals I of R such that the factor ring R/I is in \({\sigma}\). We also give a necessary and sufficient condition for an ideal \({I \neq R}\) of a ring R to be a \({\sigma}\)-ideal.

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

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 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.

Generalization of primary ideal associated with a special radical class

One of the main results of this paper is the characterization of left FP-injective rings as rings over which every finitely presented right module is torsionless. The necessity of the torsionless condition (in fact reflexivity) previously has been observed for special classes of FP-injective rings (e.g., self-injective rings (Kato and McRae), right and left coherent and self FP-injective rings (Stenström)). Furthermore, if the ring is left coherent, it is equivalent to say that every finitely presented right module is reflexive. We investigate the condition (called IF) that every injective right module is flat and characterize this via FP-injectivity. IF-rings generalize regular rings and QF-rings, and under certain chain or homological conditions coincide with regular or QF-rings (e.g., w g l dim ⁡ R ≦ 1 ⇒ regular wgl\dim R \leqq 1 \Rightarrow {\text {regular}} , noetherian, or right perfect ⇒ QF) {\text {right perfect}} \Rightarrow {\text {QF)}} .

In connection with the still unsolved problem: “Is the powerseries ring A〚T〛 over an excellent ring A excellent again?” the following question arises: “Under which assumptions is the regular locus of an universal japanese ring open?” In this paper we give conditions for the openess of regular locus for a special class of universal japanese rings.

Almost slender rings and compact rings

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.

The idea of duality occurs in many parts of mathematics. Its earliest appearance (some 150 years ago) was in projective geometry where it permitted to halve the number of theorems to be proved. It also played a most important role in analysis and topology, for example in Banach spaces and algebraic topology. In algebra, beginning with the duality of finite abelian groups as well as that of finite dimensional vector spaces, its role has been no less important. As one of the most natural generalizations of the latter for modules, Morita duality has become an important part of textbooks or a topic of lecture notes on rings and modules (cf. Anderson and Fuller [2], Cohn [23], Faith [34], Kasch [47] and Xue [84]). Since the results concerning Morita duality are very numerous and there is the excellent survey paper of Muller [70], it cannot be the purpose of this paper to make a complete report on all aspects and results relating to Morita duality. In the present survey we intend to highlight the basic ideas and observations of Morita duality (in particular, we will stress the importance of linear compactness and Grothendieck’s condition AB5*) and to show how much of this theory can be deduced from more general results about Baer duality, as well as from results of a purely category-theoretical nature. Furthermore, we will touch upon aspects of duality in some important special classes of rings, however, without aiming at completeness.

A ring P such that for every xER there exists an element c(x)ER (depending on x) such that x2c(x) -x is in the center will be called a -ring It is our purpose in the present paper to study the following conjecture of I. N. Herstein: Every -ring R is a subdirect sum of rings Ra where either (1) Ra is a division ring or (2) there exists an ideal Pa in the center of P" such that Pa = Ra or Ra/Pa

Below a special class of not necessarily associative or commutative rings A A is considered which is characterized by the property that A A has no nonzero nilpotent element and that a product of elements of A A which is equal to zero remains equal to zero no matter how its factors are associated. It is shown that ( A , ⩽ ) (A, \leqslant ) is a partially ordered set where x ⩽ y x \leqslant y if and only if x y = x 2 xy = {x^2} . Also it is shown that ( A , ⩽ ) (A, \leqslant ) is infinitely distributive, i.e., r sup x i = sup r x i r\sup {x_i} = \sup r{x_i} . Finally, based on Zorn’s lemma it is shown that A A is isomorphic to a subdirect product of not necessarily associative or commutative rings without zero divisors.