Baer Ring Research Articles

Decompositions of multiplicative semigroups of m-domain rings and reduced Rickart rings

The main result of this article is that the multiplicative semigroup of an m-domain ring is a strong semilattice of certain subsemigroups, each of which turns out to be a \rcancellative\ monoid, and that this presentation of the semigroup as a strong semilattice of \rcancellative\ semigroups is essentially unique. As a consequence, it is shown that, given an m-domain ring $ \ang{R,+,\cdot} $ with the unary operation $ \dop{} $ mapping every element to its minimal idempotent duplicator (in the sense of N.V.~Subrahmanyam), the algebra $ \ang{R,\cdot,\dop{}} $ is a strong semilattice of \rcancellative\ \dsemigroup s (in the sense of T.~Stokes), also essentially unique. Implications for reduced Rickart rings, which can be seen as a subclass of m-domain rings, are also described.

Idempotent-prime ideals and Baer-type rings with involution

In this article, we study some idempotent-structures as c P -Baer rings and I -prime rings. Moreover, we define c P -Baer *-rings and I -*-prime *-rings as involutive versions of c P -Baer rings and I -prime rings and expose their properties. Furthermore, the relation between these rings and those without involution are indicated. Finally, some extensions for c P -Baer *-rings are given, for instance the polynomial ring R [ x ] is a c P -Baer *-ring if and only if R is a c P -Baer *-ring.

Order structure of a right-strong Rickart ring under the right star order

We introduce the right star order, which is known on Rickart *-rings, in the more general setting of so-called right-strong Rickart ring s (a “star-free” generalization of Rickart *-rings). The main result is that, like a Rickart *-ring, a right-strong Rickart ring with the right star order is a relatively orthocomplemented poset. Since a right-strong Rickart ring is not necessarily involutive, this shows that the right star order can be defined and described in terms that are independent of an involution and that some of the results obtained for Rickart *-rings can be preserved.

Commutative rings whose proper ideals decompose into local modules

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

Baer and quasi-Baer annihilator conditions for nearrings and rings

A ring with unity is called Baer (quasi-Baer) if the left annihilator of each nonempty set (ideal) is generated by an idempotent element. The origins of the class of Baer rings evolved as an abstraction of the strictly algebraic properties of von Neumann algebras. This concept has been extended to nearrings. However in the classes of nearrings and rings without unity, the Baer concept splits into at least four distinct classes and at least eight classes for the quasi-Baer concept (see below). We investigate certain nearring and ring decompositions induced by Baer or quasi-Baer annihilator conditions. Examples are provided to illustrate and delimit our results.

WEAK and CO-WEAK BAER MODULES

الهدف من هذا البحث هو دراسة مفاهيم حلقات ومودولات بــــيـيـر الضعيفة وريكارت الضعيفة. لقد حصلنا على العديد من توصيفات حلقات ريكارت الضعيفة وقدمنا خصائصها. تمت دراسة العلاقة بين مودولات ريكارت الضعيفة (بيير الضعيفة) وحلقة الإندومورفيزمات الخاصة بها. لقد أثبتنا أن مودولات بيير الضعيفة التي تملك مجموعة غير منتهية من العناصر الجامدة المتعامدة غير الصفرية في حلقة الإندومورفيزمات الخاصة بها هي على وجه التحديد مودولات بيير. بالإضافة إلى ذلك، فإن حلقة الإندومورفيزمات الخاصة بمودولات ريكارت الضعيفة نصف الإسقاطية هي حلقة شبه جامدة وحلقة الإندومورفيزمات الخاصة بمودولات ريكارت المرافقة نصف الأفقية هي حلقة شبه جامدة. علاوة على ذلك، فقد بينا أن المودولات الحرة هي مودولات بيير الضعيفة إذا وفقط إذا كانت حلقة الإندومورفيزمات الخاصة بها هي حلقة بيير ضعيفة يسارية.

On existence of joins and meets under the star order in strong Rickart rings

A strong Rickart ring is a Rickart ring which has some special properties that make it similar, in some respects, to Rickart *-rings. It is known that the star partial order on a Rickart *-ring can be characterized by a condition not involving involution; this allows one to define the star order also on strong Rickart rings. In the paper, we reconsider several conditions, known in the literature, for existence of joins and meets in Rickart *-rings under the star order. Our main purpose is to transfer the conditions, even those involving involution, to strong Rickart rings and to show that they are valid also in this wider class of rings.

$\pi$-BAER $\ast$-RINGS

A $\ast$-ring
 $R$ is called a $\pi$-Baer $\ast$-ring, if for any projection invariant left ideal $Y$ of $R$, the right annihilator of $Y $
 is generated, as a right ideal, by a projection.
 In this note, we
 study some properties of such $\ast$-rings.
 We indicate interrelationships between the $\pi$-Baer $\ast$-rings and related classes of rings such as
 $\pi$-Baer rings, Baer $\ast$-rings, and quasi-Baer $\ast$-rings. We announce several
 results on $\pi$-Baer $\ast$-rings.
 We show that this notion is well-behaved with respect to
 polynomial extensions and full matrix rings.
 Examples are provided to explain and delimit our results.

When a (dual-)Baer module is a direct sum of (co-)prime modules

Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module M is a direct sum of prime modules, then every direct summand of M is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of M is finite). Dually, if every direct summand of a dual-Baer module M is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or M is a max-module. Among other applications, we show that if R is a commutative hereditary Noetherian ring then a finitely generated R-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.

Rings which are Baer or quasi-Baer modulo a radical

Baer and quasi-Baer rings are important classes of algebraic objects, and their properties have roots in analysis. In this paper, we investigate rings R such that is Baer or quasi-Baer, where is either the Jacobson radical or the prime radical of R. Preliminary characterizations and results are obtained; in particular, we show that the property of being quasi-Baer is a Morita invariant.

On the lattice of annihilator ideals and its applications

In this paper we study the properties of (the set of right annihilator ideals of R) as a lattice. We prove that is a Boolean algebra if and only if R is a semiprime ring. For a quasi Armandariz ring R, we show that the lattices and are isomorphic. In general, for any ring R, we show that the lattices and are isomorphic. We answer two questions related to property (a.c.) that were raised by Hong, Kim, Lee and Nielsen. To do this, we introduce a new class of rings. We say R is right strongly quasi Armandariz (-Armandariz) if for each there exists such that The class of right -Armandariz rings is between right -Baer rings and Armandariz rings. For a right -Armandariz ring R, it is proved that R has the right annihilator condition (a.c.) if and only if R[x] has right the annihilator condition (a.c.). We conclude that if R is a quasi Armandariz ring and has right (a.c.), then R[x] has right (a.c.).

On Projection Invariant Semisimple Modules

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties of aforementioned class of modules are studied. We obtain indecomposable decompositions of former class of modules under some module theoretical conditions. Moreover, we explore when the finite exchange property implies full exchange property for the class of projection invariant semisimple modules. Finally, we obtain that the endomorphism ring of a projection invariant semisimple modules is a π- Baer ring.

On Projection Invariant Semisimple Modules

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties of aforementioned class of modules are studied. We obtain indecomposable decompositions of former class of modules under some module theoretical conditions. Moreover, we explore when the finite exchange property implies full exchange property for the class of projection invariant semisimple modules. Finally, we obtain that the endomorphism ring of a projection invariant semisimple modules is a π- Baer ring.

On dual $\pi$-endo Baer modules

We introduce the concept of dual $\pi$-endo Baer modules. We evolve several structural properties such as direct summands and direct sums. Moreover, we prove that the endomorphism ring of a dual $\pi$-endo Baer module is a $\pi$-Baer ring. The examples are presented to exhibit the results.

Some results on relative dual Baer property

Let $R$ be a ring.
 In this article, we introduce and study relative dual Baer property.
 We characterize $R$-modules $M$ which are $R_R$-dual Baer, where $R$ is a commutative principal ideal domain.
 It is shown that over a right noetherian right hereditary ring $R$, an $R$-module $M$ is $N$-dual Baer for
 all $R$-modules $N$ if and only if $M$ is an injective $R$-module.
 It is also shown that for $R$-modules $M_1$, $M_2$, $\ldots$, $M_n$ such that $M_i$ is $M_j$-projective for all
 $i > j \in \{1,2,\ldots, n\}$, an $R$-module $N$ is $\bigoplus_{i=1}^nM_i$-dual Baer if and only if $N$ is
 $M_i$-dual Baer for all $i\in \{1,2,\ldots,n\}$.
 We prove that an $R$-module $M$ is dual Baer if and only if $S=End_R(M)$ is a Baer ring
 and $IM=r_M(l_S(IM))$ for every right ideal $I$ of $S$.

Skew inverse Laurent series extensions of weakly principally quasi Baer rings

A ring [Formula: see text] is called weakly principally quasi Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left [Formula: see text]-unital by left semicentral idempotents, which implies that [Formula: see text] modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly p.q.-Baer property of a ring [Formula: see text] and those of the skew inverse series rings [Formula: see text] and [Formula: see text], for any automorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text] of [Formula: see text]. Examples to illustrate and delimit the theory are provided.

The reduced ring order and lower semi-lattices II

This paper continues the study of the reduced ring order (rr-order) in reduced rings where [Formula: see text] if [Formula: see text]. A reduced ring is called rr-good if it is a lower semi-lattice in the order. Examples include weakly Baer rings (wB or PP-rings) but many more. Localizations are examined relating to this order as well as the Pierce sheaf. Liftings of rr-orthogonal sets over surjections of reduced rings are studied. A known result about commutative power series rings over wB rings is extended, via methods developed here, to very general, not necessarily commutative, power series rings defined by an ordered monoid, showing that they are wB.

Minus partial order in regular modules

The minus partial order is already known for sets of matrices over a field and bounded linear operators on arbitrary Hilbert spaces. Recently, this partial order has been studied on Rickart rings. In this paper, we extend the concept of the minus relation to the module theoretic setting and prove that this relation is a partial order when the module is regular. Moreover, various characterizations of the minus partial order in regular modules are presented and some well-known results are also generalized.

Rickart gamma rings

We introduce the concepts of Rickart gamma ring as a generalized of a Rickart ring and Baer gamma ring. Also, For an Γ —ring R, we show that: (1) R is a right Rickart if and only if it is right p.p.. (2) R is a Baer, quasi-Baer if and only if R is Rickart, p.q. Baer (respectively) and R = {eΓR|e 2 = e ∈ R|, under inclusion, is complete lattice. (3) R is prime if and only if it is quasi-(or right p.q.) Baer and semicentral reduced. (4) If R is Rickart then: (a) R is both right and left nonsingular. (b) R has no nonzero central nilpotent elements. (c) The image isomorphic of R is Rickart. (d) If R is reduced then the idempotent element which is generated the right annihilator of any element in is R unique. (5) The direct product Π i∈I Ri of Γ —rings is Rickart, quasi-(right p.q. ) Baer iff Ri is Rickart quasi-(right p.q.) Baer for all i ∈ I, respectively. (6) The corner and the center of Quasi- Baer, p.q. Baer and Rickart are Quasi-Baer and Baer, p.q. and Rickart and, by conditions, Rickart and Rickart, respectively. (7) We give some conditions to show that (a) If R is Rickart, Quasi-Baer then R Baer and p.q.- Baer, Rickart, respectively. (b) If R is Rickart, then the following statements of R are equivalent: reduced, abelian, idempotents of R commute, the set of idempotents is closed under multiplication, R is commutative at 0, RFI(x) = LFI(x) for every x ∈ R and rR (x) = lR (x) for all x ∈ R.

Correction to Notes on reduced Rickart rings, I. Representation and equational axiomatizations

Corrected are some misprints and Theorem 6.5 in the joint paper with I. Cremer, “Notes on reduced Rickart rings, I. Representation and equational axiomatizations”.