Regular self-injective rings andV-rings

Regular, right self-injective rings are considered. We settle the question of when such a ring is a right V-ring, i.e., when each simple right module over the ring is injective. It is proved that a regular, right self-injective V-ring of the power of the continuum has a bounded nilpotency index.Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 564–575, September–October, 1994.     

Regular rings and distributive lattices

We show that a finite lattice is isomorphic to the lattice of ideals in a regular ring if and only if it is distributive. We also study the countable case.     

Regular odd rings and non-planar graphs

In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Representation of artinian partially ordered sets over semiartinian von Neuman regular algebras

If R is a semiartinian von Neumann regular ring, then the set ${\mathbf{Prim}}_R$ of primitive ideals of R, ordered by inclusion, is an artinian poset in which all maximal chains have a greatest element. Moreover, if ${\mathbf{Prim}}_R$ has no infinite antichains, then the lattice ${\mathbb{L}}_2(R)$ of all ideals of R is anti-isomorphic to the lattice of all upper subsets of ${\mathbf{Prim}}_R$. Since the assignment $U?r_R(U)$ defines a bijection from any set ${\mathbf{Simp}}_R$ of representatives of simple right R-modules to ${\mathbf{Prim}}_R$, a natural partial order is induced in ${\mathbf{Simp}}_R$, under which the maximal elements are precisely those simple right R-modules which are finite dimensional over the respective endomorphism division rings; these are always R-injective. Given any artinian poset I with at least two elements and having a finite cofinal subset, a lower subset $I^{\prime }?I$ and a field D, we present a construction which produces a semiartinian and unit-regular D-algebra $D_I$ having the following features: (a) ${\mathbf{Simp}}_{D_I}$ is order isomorphic to I; (b) the assignment $H?{\mathbf{Simp}}_{D_I/H}$ realizes an anti-isomorphism from the lattice ${\mathbb{L}}_2(D_I)$ to the lattice of all upper subsets of ${\mathbf{Simp}}_{D_I}$; (c) a non-maximal element of ${\mathbf{Simp}}_{D_I}$ is injective if and only if it corresponds to an element of $I^{\prime }$, thus $D_I$ is a right V-ring if and only if $I^{\prime }=I$; (d) $D_I$ is a right and left V-ring if and only if I is an antichain; (e) if I has finite dual Krull length, then $D_I$ is (right and left) hereditary; (f) if I is at most countable and $I^{\prime }=?$, then $D_I$ is a countably dimensional D-algebra.     

Regular Homomorphisms and Regular Maps

Regular homomorphisms of oriented maps essentially arise from a factorization by a subgroup of automorphisms. This kind of map homomorphism is studied in detail, and generalized to the case when the induced homomorphism of the underlying graphs is not valency preserving. Reconstruction is treated by means of voltage assignments on angles, a natural extension of the common assignments on darts. Lifting and projecting groups of automorphisms along regular homomorphisms is studied in some detail. Finally, the split-extension structure of lifted groups is analysed.     

正则关系与完全正则空间

借助于关系的某些代数性质刻画拓扑空间的完全正则性,证明了拓扑空间是完全正则的当且仅当其闭集与开集之间存在满足一定简单条件的正则关系,有限正则关系或广义有限正则关系.     

带正则*-断面的正则半群

本文给出了带正则*-断面的正则半群的若干性质,获得了带拟理想正则*-断面的正则半群的一个构造方法.利用这一构造定理,考虑了这类半群上的同余.     

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.