Efraim P. Armendariz

A note on extensions of Baer and P. P. -rings

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R [X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jondrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

Central localizations of regular rings

In this paper we show that a ring R is von Neumann regular (or a V-ring) if and only if every central localization of R at a maximal ideal of its center is von Neumann regular (or a V-ring). Strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. Also we consider whether regular PI-rings can be characterized by the property that all central localizations at maximal ideals of the center are simple. Commutative von Neumann regular rings have been characterized in various ways. However, very few of these characterizations extend to noncommutative rings. The results in this paper arose from attempting to extend to noncommutative rings a well-known theorem of Kaplansky [11, Theorem 6] which states that commutative regular rings are characterized by the property that all localizations at maximal ideals are fields. It turns out that the obvious extension of this theorem to the noncommutative case is valid. That being, a ring is regular if and only. if all central localizations at maximal ideals of the center are regular. An analogous theorem is obtained for Vrings. Also we show that strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. With an eye to the commutative theory, we consider whether regular PIrings can be characterized by the property that all central localizations at maximal ideals of the center are simple. We provide an example to show that this is not the case. However, it is true if and only if contraction provides a 1:1 correspondence between maximal ideals of the ring and maximal ideals

RINGS CONTAINING IDEALS WITH BOUNDED INDEX

Throughout R will denote an associative ring not necessarily with unity. Recall that a ring R is said to have bounded index if there is a positive integer n 1 such that x 1⁄4 0 whenever x is a nilpotent element of R. The least such positive integer is called the index of R, and we denote it by iðRÞ. A ring R is said to be reduced if iðRÞ 1⁄4 1 (i.e., R has no nonzero nilpotent elements). For example, any semiprime PI-ring has bounded index [1, Theorem 10.8.2] and any semiprime right Goldie ring has bounded index. By [2, Proposition 4], any semiprime ring with bounded index is right (and left) nonsingular. We say that an ideal of R (resp. right ideal of R) A is essential (resp. right essential) in R if A has nonzero intersection with every nonzero ideal

The double centralizer theorem for division algebras

AssumeV is a finite-dimensional vector space over a division ringD having centerF. It is shown that ifT∈ EndD(V) is algebraic overF then the double centralizerC(C(T)) ofT is the setF[T] of all polynomials inT with coefficients fromF. Consequently, eachn×n matrix ring overD is an algebraicF-algebra if and only ifC(C(T))=F[T] for allT and all finite-dimensionalV.

Baer rings satisfying J2 = J For all ideals

A ring satisfies J2 = J for all ideals J if and only if all of its homomorphic images are semiprime; we call such rings ‘fully semiprime’. D. Castellas theory of compressibility of regular Baer rings is extended to fully semiprime Baer rings and Baer ∗—rings. Sample application: If a fully semiprime Baer ring is a matrix ring, then its center is self-injective

On prime ideals in hereditary PI-rings

Abstract It is shown that if P is a prime ideal of a right hereditary ring R satisfying a polynomial identity, then R P is a hereditary Noetherian ring.

RESTRICTED SEMIPRIMARY RINGS

This chapter describes restricted semiprimary rings. It discusses the concept of a commutative ring all of whose factors are Artinian. It is found that as the class of semiprimary rings extends the class of Artinian rings, RSP-rings encompass RM-rings. It is observed that for commutative RSP-domains, integral extensions are RSP-domains and torsion modules have the primary decomposition property. For a commutative RSP-domain A , the torsion A -modules are precisely the A -modules having a socle sequence. The class of finitely generated submodules of N containing A as a dense submodule is closed under finite sums, hence directed as a partially ordered set, it has ascending chain condition, and it has a maximal member B . It is found that if C is a finitely generated submodule of N containing B , then R is left semihereditary. The result of Bass asserts that the quotient of R by its Jacobson radical J will be semisimple Artinian.