Elementary regular rings

Translated fromAlgebra i Logika, Vol. 32, No. 4, pp. 387–401, July–August, 1993.     

On regular rings and self-injective rings

This note contains certain conditions for unit-regularity and for selfinjective regular rings to have bounded index (such rings have many interesting properties [9]). Sufficient conditions are also given for rings to be Noetherian, Artinian and quasi-Frobeniusian.     

Morphic rings and unit regular rings

A ring R is called left morphic if for every a∈R. A left and right morphic ring is called a morphic ring. If M_n(R) is morphic for all n≥1 then R is called a strongly morphic ring. A well-known result of Erlich says that a ring R is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring R is unit regular iff R[x]/(xⁿ) is strongly morphic for all n≥1 iff R[x]/(x²) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains. This places some known examples in a broader context and answers some existing questions.     

Twisted bounded-dilation group C*-algebras as C*-metric algebras

We construct a class of C~*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ,the closure of the seminorm ||[M?, ·]|| on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C~*-algebra C*r(Γ, σ) for the pointwise multiplication operator M?on ?2(Γ), induced by a proper length function ? on Γ with the property of bounded θ-dilation. Moreover, the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.     

On regular semigroup rings

Communicated by L. N. Shevrin     

Finiteness in regular rings

Translated from Algebra i Logika, Vol. 31, No. 2, pp. 189–197, March–April, 1992.     

On regular rings and continuous rings,III

Summary In this sequel to [14]and [15],a generalization of quasi-injective modules, noted FK-injective, is introduced to study von Neumann regular and continuous rings. This will lead to new characteristic properties of continuous regular rings. Conditions for certain non-singular modules to be completely reducible and injective are given. A few decompositions of FK-injective modules are considered.Dedicated to Professor Hisao Tominaga     

Approximating discrete valuation rings by regular local rings

Let be a field of characteristic zero and let be a discrete rank-one valuation domain containing with . Assume that the fraction field of has finite transcendence degree over . For every positive integer , we prove that can be realized as a directed union of regular local -subalgebras of of dimension .

     

Prime regular rings of quotients

Let R be a ring with 1, and Q its maximal ring of right quotients. We give a number of necessary and sufficient conditions on R so that q is prime regular. We obtain as corollaries conditions so that Q is a full linear ring or Q is simple and satisfies xy = 1 implies yx = 1.     

Strongly regular rings and rational identities of division rings

We look at division rings in the variety of strongly regular rings and show a connection to the study of rational identities on division rings.Research partially supported by a Grant from NSERC.