On a Theorem of Burgess and Stephenson

A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.

Left Rings

We introduce in this paper the concept of left rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left rings can be extended to left rings. As applications, we are able to give some new characterizations of regular left self-injective rings with nonzero socle and extend some known results on strongly regular rings.

On the left strongly prime modules and their radicals

We give the new results on the theory of the one-sided (left) strongly prime modules and their strongly prime radicals. Particularly, the conceptually new and short proof of the A.L.Rosenberg’s theorem about one-sided strongly prime radical of the ring is given. Main results of the paper are: presentation of each left stongly prime ideal p of a ring R as p = R ∩ M, where M is a maximal left ideal in a ring of polynomials over the ring R; characterization of the primeless modules and characterization of the left strongly prime radical of a finitely generated module M in terms of the Jacobson radicals of modules of polynomes M(X1, . . . , Xni) .

MC2 RINGS AND WQD RINGS

We introduce in this paper the concepts of rings characterized by minimal one-sided ideals and concern ourselves with rings containing an injective maximal left ideal. Some known results for idempotent reflexive rings and left HI rings can be extended to left MC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with non-zero socle and extend some known results for strongly regular rings.

The structure of a subclass of amenable banach algebras

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian*-algebras that are contractible.

Injective endomorphisms and maximal left ideals of left Artinian rings

Given a ring R and an injective ring endomorphism α: R → R, not necessarily surjective, it is possible to define a minimal overring A(R, α) of R to which extends as an automorphism. The ring A(R, α) was first studied by D. A. Jordan in his paper [5], where he also introduces the central objects of this paper—the closed left ideals of R. As can be seen from Theorem 4.7 of [5], the left ideal structure of A(R, α) depends very strongly on the closed left ideals of R, and our aim here is to show that each maximal left ideal of a left Artinian ring is closed.

Maximal Left Ideals and Idealizers in Matrix Rings

Let R be a ring with identity, Mn(R) the ring of n × n matrices over R. The lattice of two-sided ideals of R is carried via A → Mn(A) to form the lattice of two-sided ideals of Mn(R). We wish to study the more complex left ideal structure of Mn(R). For example, if K is a commutative field, then Mn(K) has non-trivial left ideals. In particular Mn(K) has the maximal left ideal consisting of all matrices with some designated column zero. Or for any ring with maximal left ideal M, Mn(R) has the maximal left ideal consisting of all matrices with some column's entries from M. In Theorem 1.2 we characterize the maximal left ideals of Mn(R) in terms of those of R. We briefly study some contraction properties of maximal left ideals in matrix rings. For R commutative we “count” the maximal left ideals of Mn(R) and describe the idealizer of any such ideal; in the case where K is a field we see that the collection of maximal left ideals of Mn(K) can be naturally identified with Pn–1(K) (projective space).