On q-Galois Extensions of Simple Rings

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

A Galois theory of commutative rings

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.

In 1996, the first author defined analogs of the concepts of complete (divisible), reduced, and periodic abelian groups, well-known in the theory of abelian groups, for arbitrary varieties of algebras. In 2021, the first author proposed a modification of the concepts of completeness and reducibility, which is more natural in the case of associative rings. The paper studies the modification of these concepts for associative rings. Artinian \(\mathbf{M}\)-complete, \(\mathbf{M}\)-reduced rings, and minimally \(\mathbf{M}\)-complete associative nilpotent rings, simple rings with unity, and finite rings are characterized.

Introduction.The author [5] has formulated sufficient conditions on a ring Ti and a group G of automorphisms of Ti to derive a Galois theory of noncommutative rings which extends the Galois theory of commutative rings developed by Chase, Harrison, and Rosenberg [3].This paper continues the study ofthat Galois theory, investigating the structure of the lattice of left ideals in Ti and the existence of normal bases for Ti.

In this chapter we shall analyze arbitrary extension fields of a field Φ. A study of finite dimensional extension fields and a partial study of algebraic extensions has been made in Chapter I. In this chapter our primary concern will be with infinite dimensional extensions and we shall begin again with the algebraic ones. We define algebraically closed fields and prove the existence of an algebraic closure of any field. We shall extend the classical Galois theory to apply to infinite dimensional normal and separable extensions. After this we shall consider arbitrary extension fields and we shall show that these can be built up in two stages: first a purely transcendental one and then on top of this an algebraic extension. The invariant of this mode of generating a field is the transcendency degree which is the cardinal number of a transcendency basis. We shall obtain conditions for the existence of a transcendency basis such that the extension is separable algebraic over the purely transcendental extension determined by the basis. We shall also give a definition of separability of an extension field that generalizes the notion of algebraic separability. The notion of a derivation plays an important role in these considerations.KeywordsPrime IdealStructure TheoryGalois GroupHigh DerivationAlgebraic ClosureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Herstein's Lie theory revisited

The third chapter contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface. For such surfaces, the geometry of ramified coverings and Galois theory are not only analogous but in fact very closely related to each other. This relationship is useful in both directions. On the one hand, Galois theory and Riemann’s existence theorem allow one to describe the field of functions on a ramified covering over a Riemann surface as a finite algebraic extension of the field of meromorphic functions on the Riemann surface. On the other hand, the geometry of ramified coverings together with Riemann’s existence theorem allows one to give a transparent description of algebraic extensions of the field of meromorphic functions over a Riemann surface.

Recently, the study of probabilities in ring theory has shown a significant increase in the field of algebra. Many interesting algebraic structures were modeled to find their probabilities in certain finite rings. In this paper, we introduce a new type of probability in finite rings, namely the squared-zero product probability. The aim is to study the square property and the zero product property of the ring. The focus of this study is the ring of matrices of dimension two over integers modulo prime p. To obtain the probability, the order of the square-annihilator of the ring is determined. The results found show that the squared-zero product probability of the ring is dependent on the value of p.

Wedderburn’s theorem entirely reduces the study of semisimple algebras of finite rank over a field K to that of division algebras of finite rank over the same field. We now concentrate on this problem. If D is a division algebra of finite rank over K and L the centre of D then L is a finite extension of K and we can consider D as an algebra over L. Hence the problem divides into two: to study finite field extensions, which is a question of commutative algebra or Galois theory, and that of division algebras of finite rank over a field which is its centre. If an algebra D of finite rank over a field K has K as its centre, then we say that D is a central algebra over K.KeywordsNumber FieldDivision AlgebraIsosceles TriangleFinite RankSimple Central AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Brauer group Br(R) of a commutative ring was introduced by Auslander and Goldman in their 1960 paper The Brauer Group of a Commutative Ring, building on earlier work of Azumaya. This group coincides with the “classical” Brauer group (cf. Chapter 4) in the case when R is a field. One of the points of extending the theory to rings is that one can relate Brauer groups of fields to Brauer groups of related rings in exact sequences; one then hopes that this will help compute the classical Brauer group. The Brauer group of a commutative ring is also part of a Galois theory of commutative rings. For more on these matters, the reader may consult Galois Theory and Cohomology of Commutative Rings by Chase, Harrison and Rosenberg, The Brauer Group of Commutative Rings by Orzech and Small, Separable Algebras Over Commutative Rings by DeMeyer and Ingraham, or the paper of Auslander and Goldman quoted above.

ATTENTION has been given in NATURE to various deductions from the results of Dr. P. E. Shaw's experiments on The Newtonian Constant of Gravitation as affected by Temperature (Phil. Trans., A, 544, 1916). So far as the present writer is aware, attention has not been directed to the suggestive remarks of the late Prof. G. F. Fitzgerald, to be found in his Helmholtz memorial lecture (Transactions Chemical Society, 1896, pp. 889–95). In the course of his reference to Helmholtz's contribution to the theory of vortex motion, Fitzgerald remarks:—It is difficult to weigh hot bodies accurately, and, in consequence, there does not seem to be any conclusive proof that the weight of a body does not change with its temperature. If it does not do so by a measurable amount, the simple vortex ring theory of matter can hardly be true.

In 1942 Artin proved the linear independence, over a field S, of distinct automorphism of S; in other words if G is a finite group of automorphisms of S and R is the fixed field, then Horn^S, S) is a free S-module with G as basis. Since then, this last condition (“S is G-Galois”) or its equivalents have been used as a postulate in all the Galois theories of rings that are not fields, for example by Dieudonné, Jacobson, Azumaya and Nakayama for noncommutative rings and then in [AG, Appendix] and [CUR] for commutative rings. When S has no idempotents but 0 and 1, [CHR] proves that the ordinary fundamental theorem of Galois theory holds with no real change from the classical, field case.

Generalized Koszul algebra and Koszul duality

A generalization of the Katzman-Zhang algorithm