Associative rings

This survey mainly covers material abstracted in theReferativnyi Zhurnal Matematika during the period 1977–1983 and is a continuation of earlier surveys on the same subject, the theory of associative rings. An extensive bibliography of 1201 titles is included.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 22, pp. 3–115, 1984.     

Associative rings with large center

A ring R is called a ring with large center if any nonzero ideal of R has nonzero intersection with the center of R. We give some conditions for an ideal of a ring with large center to be itself a ring with large center, and also we provide an example of a ring with large center R and its ideal I ? R such that I is not a ring with large center.     

Associative rings satisfying the Engel condition

Let be a commutative ring, and let be an associative -algebra generated by elements . We show that if satisfies the Engel condition of degree , then is upper Lie nilpotent of class bounded by a function that depends only on and . We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.

     

Associative rings the radicals over which are subrings

The paper studies the structure of algebras which are radicals over PI-subalgebras. In particular, a theorem is proven to the effect that an algebra without nil-ideals which is a radical over a right Pi-ideal is a Pi-algebra.Translated from Matematicheskie Zametki, Vol.11, No.1, pp. 33–40, January, 1972.In conclusion, the author wishes to thank L. A. Bokut for help with and interest in this paper.     

关于内结合环的注记（英）

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Free Associative Algebras

On page 37, lines 7–8, for "principal ideal domain" read"Euclidean domain".     

Weakly Associative Groups

The space ? _c ³ of 3-dimensional relativistically admissible velocities possesses (i) a binary operation which represents the relativistic velocity composition law; and (ii) a mapping from the cartesian product ? _c ³ ×? _c ³ into a subgroup of its automorphism group, Aut(? _c ³ ), representing the Thomas precession of special relativity. These binary operation and mapping are studied in special relativity as two isolated phenomena. It was recently discovered, however, that they are linked by an algebraic structure which gives rise to a theory of weakly associative and weakly associative-commutative groups. The axioms of these groups are presented in this paper and employed to obtain various interesting results. The algebraic structure underlying these nonstandard groups has been discovered and studied in a totally different context by Karzel (1965), Kerby and Wefelscheid.     

Associative n-Tuple algebras

In the paper,we study algebras having n bilinearmultiplication operations : A×A → A, s = 1, …, n, such that (a b) c = a (b c), s, r = 1,..., n, a, b, c ∈ A. The radical of such an algebra is defined as the intersection of the annihilators of irreducible A-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is s-regular for some operation . This implies that the quotient algebra by the radical is semisimple. If an n-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian n-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional n-tuple algebra A, over an algebraically closed field, which is a simple A-module.     

An Algebra Which Is Power Associative but not Strictly Power Associative

We construct a power associative algebra over the field with 3 elements which is no longer power associative when the scalars are extended to the field with 9 elements.     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.     

Local rings of exchange rings

We characterize the exchange property for non-unital rings in terms of their local rings at elements,and we use this characterization to show that the exchange property is Morita invariant for idempotent rings.We also prove that every ring contains a greatest exchange idela(with respect to the inclusion).     

Branch rings, thinned rings, tree enveloping rings

We develop the theory of “branch algebras”, which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field % MathType!End!2!1! we contruct a % MathType!End!2!1! which

Quaternion rings and octonion rings

In this paper, for rings R, we introduce complex rings ?(R), quaternion rings ?(R), and octonion rings O(R), which are extension rings of R; R ? ?(R) ? ?(R) ? O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ?(R) and ?(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.     

约化环和半交换环

讨论了在约化条件下,比平凡扩张更广泛的一类扩张环的半交换性.通过给出半交换模的定义,得到平凡扩张是半交换环的一个充要条件.