Self-injective Regular Ring Research Articles

Automorphism-invariant non-singular rings and modules

A ring A is a right automorphism-invariant right non-singular ring if and only if A=S×T, where S a right self-injective regular ring and T is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. Over a ring A, each direct sum of automorphism-invariant non-singular right modules is an automorphism-invariant module if and only if the factor ring of the ring A with respect to its right Goldie radical is a semiprime right Goldie ring.

Rings whose cyclics are C3-modules

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

C-injectivity and C-projectivity

The definition of p-injectivity motivates us to generalize the notion of injectivity and projectivity, noted respectively C-injectivity and C-projectivity. Noetherian, semi-simple Artinian, quasi-Frobenius, regular hereditary and self-injective regular rings are considered in terms of C-injectivity and C-projectivity. Partial answers are given to Matlis’ Problem and Boyle’s Conjecture.

Centers and Two-Sided Ideals of Right Self-Injective Regular Rings

In this paper we prove two main results. The first one is the existence of a right self-injective regular ring of type III with center any given commutative self-injective regular ring. The second one is a characterization of the lattice of two-sided ideals for a large class of right self-injective regular rings of type III. In a previous paper, this lattice was described as the lattice of order ideals of a certain lattice of continuous functions. Now we prove the converse: for a large class of complete boolean spaces X, let C be the set of all continuous functions from X into {0} ∪ [ℵ0, γ], where γ is any infinite cardinal number, and let Δ be the lattice of all functions of C that are less than or equal to a given nonvanishing function of C. Then there exists a right self-injective regular ring R of type III such that the lattice of two-sided ideals of R is isomorphic to the lattice of order ideals of Δ.

On the structure of selfinjective regular rings

On the structure of selfinjective regular rings

On existence of directly finite, only one sided self-injective regular rings

This paper is concerned with the following open problem. Is every directly finite, regular right self-injective ring necessarily left self-injective [2, Open problem 14]? This problem is 21 years old and came from J.-E. Roos [8]. We solve in the negative by giving an example. In the example, we construct a simple regular subring of the rank metric completion of a direct limit of a sequence of matrix rings over a field. Moreover the maximal right quotient ring of this simple regular ring is directly finite and not left self-injective. This result was announced in [6] without proof.

Two-sided ideals in right self-injective regular rings

For a large class of purely infinite right self-injective regular rings, wedescribe the lattice of two-sided ideals as a lattice of ideals of a certain lattice of continuous functions. For rings of type I or II, our result is a generalization of the corresponding one in the prime case obtained by Goodearl. The main tools that we use are the relative and infinite Goodearl-Boyle dimension functions. These functions are glued together in order to obtain a dimension function D defined on the lattice of principal right ideals. The computation of the range of D constitutes the key point in proving our main result.

Centres of Rank-Metric Completions

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

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.

Perspectivity and cancellation in regular rings

Let R be a (von Neumann) regular ring, and L(R) the compIemented modular lattice of principal right ideals of R. In [6], it is shown that for suitable regular rings, and suitable modules A, B, C, z4@BzA@C implies B N C. This is a type of cancellation law that works e.g., for finitely generated projective modules over a “finite” self-injective regular ring. It turns out that for finitely generated projective modules over regular rings this cancellation law holds precisely when the ring is unit regular [definitions follow). A related problem is the transitivity of perspectivity on L(R). We show that transitivity holds on a two by two matrix ring over R if and only if R is unit regular. An example of Bergman’s is included to show that transitivity on L(R) itself is not sufficient for unit regularity. Along the way, we obtain the following result for arbitrary complemented lattices: If perspectivity is cancellative (in the sense above), it is transitive; if perspectivity is transitive, it is additive (in the obvious sense). Another example, also due to Bergman, is included. I am also indebted to George Bergman for many ideas and results, in particular the concept of a von Neumann module. All rings are associative and possess I. A ring is regular if all finitely generated right (or left) modules are generated by an idempotent. Equivalently, every element a of R possesses a quasi-inverse; that is, an element h with aba = b. The collection of principal right ideals is then a complemented modular lattice, which we denote by L(R). We use either of two notations: eR (for e = es in R) or JEL(R) to denote a principal right ideal If

Partially Self-Injective Regular Rings

AbstractIt is proved, for any uncountable cardinal λ, that a λ-complete Boolean ring is λ-self-injective. An example shows that the converse need not hold.