Unit-regular Rings Research Articles

A new member of Nicholson's morphic folks

The main aim of the paper is to describe the structure of modules and corresponding rings satisfying the property (P), which says that M/im(α) is embeddable into ker⁡(α) for each endomorphism α and which generalizes the morphic property. In particular, it is proved that the class of rings with the property (P) is closed under taking products and summands and contains unit regular rings. We also explain connections between the virtually internal cancellation property and the property (P) and characterize the structure of particular classes of rings satisfying the property (P).

On (quasi-)morphic property of skew polynomial rings

The main objective of this paper is to study (quasi-)morphic property of skew polynomial rings. Let $R$ be a ring, $\sigma$ be a ring homomorphism on $R$ and $n\geq 1$. We show that $R$ inherits the quasi-morphic property from $R[x;\sigma]/(x^{n+1})$. It is also proved that the morphic property over $R[x;\sigma]/(x^{n+1})$ implies that $R$ is a regular ring. Moreover, we characterize a unit-regular ring $R$ via the morphic property of $R[x;\sigma]/(x^{n+1})$. We also investigate the relationship between strongly regular rings and centrally morphic rings. For instance, we show that for a domain $R$, $R[x;\sigma]/(x^{n+1})$ is (left) centrally morphic if and only if $R$ is a division ring and $\sigma(r)=u^{-1}ru$ for some $u\in R$. Examples which delimit and illustrate our results are provided.

The generalized Flanders’ theorem in unit-regular rings

Let R be a unit-regular ring, and let a, b, c ? R satisfy aba = aca. If ac or ba is Drazin invertible, we prove that their Drazin inverses are similar. Furthermore, if ac and ba are group invertible, then ac is similar to ba. For any n?n complex matrices A, B,C with ABA = ACA, we prove that AC and BA are similar if and only if their k-powers have the same rank. These generalize the known Flanders? theorem proved by Hartwig.

Perspectivity and von Neumann regularity

We investigate connections between von Neumann regularity of endomorphisms and perspectivity of direct summands in modules. This leads to a new classification of those rings whose regular elements are strongly regular, which turn out to be exactly the rings R whose idempotents are central modulo the Jacobson radical J(R). An important component of our work is an investigation of the left and right associate relations on idempotents, as well as chains of these relations. As applications we give new characterizations of strongly regular elements and of idempotents that are central modulo the Jacobson radical. We also introduce a new class of regular elements that we call pc-regular elements, related to perspectivity in complement summands. These pc-regular elements are exactly the special clean elements. Generalizing the well-known fact that unit-regular rings are special clean, we then show that the unit-regular elements of any regular ring satisfying general comparability are special clean. Consequently, unit-regular endomorphisms of quasi-continuous modules are special clean, answering, in the positive, a conjecture of Lam.

Generalizing unit-regular rings and special clean elements

As a strengthening of the definition of weakly clean rings, given by Ster in J. Algebra (2014), and as a common generalization of the classical unit-regular rings, we define and investigate the class of socalled weakly unit-regular rings as those rings R for which, for every element a ? R, there exist a unit u and an idempotent e such that a ? u ? e ? (1 ? e)Ra with aR ? eR = {0}. Some more exotic relationships with the well-known classes of clean, nil-clean and (strongly) ?-regular rings are demonstrated as well. In particular, an elementwise extension of the so-called ”special clean elements” by Khurana et al. in J. Algebra & Appl. (2020) is also processed.

Unit Regular Clean Rings

A ring R is called unit regular clean, if every element is the sum of an idempotent and a unit regular elements. In this paper we introduce the notion of unit regular clean ring. we investigate some of it’s basic properties and it’s relation with clean ring.

RINGS WITH THE DUAL OF THE ISOMORPHISM THEOREM

A ring R satisfies the dual of the isomorphism theorem if R/Ra≅ 1(a) for all elements a of R, where 1(a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings.

On (Unit-)Regular Morphisms

We introduce a symmetry property for unit-regular rings as follows: a ∈ R is unit-regular if and only if aR ⊕ (a − u)R = R (equivalently, Ra ⊕ R(a − u) = R) for some unit u of R if and only if aR ⊕ (a − u)R =(2a − u)R (equivalently, Ra ⊕ R(a − u) = R(2a − u)) for some unit u of R. Let M and N be right R-modules and α, β ∈ Hom(M, N) such that α + β is regular. It is shown that αS ⊕ βS =(α + β)S, where S = End(M) if and only if Tα ⊕ Tβ = T(α + β), where T = End(N). We also introduce partial order α ≤⊕β and minus partial order α ≤−β for any α, β ∈ Hom(M, N); they translate into module-theoretic language defined in a ring in [7] and [8]. We analyze some relationships between ≤⊕ and ≤− on the endomorphism rings of the modules M and N.

Characterizations and Direct Sums of Unit-Endoregular Modules

AbstractA module is called unit-endoregular if its endomorphism ring is unit-regular. In this paper, we continue the research in unit-endoregular modules. More characterizations of unit-endoregular modules are obtained. As a special case, we show that for an abelian group G, Endℤ(G) is a unit-regular Baer ring if and only if Endℤ(G) is a two-sided extending regular ring. While the class of unit-endoregular modules is not closed under direct sums, we provide a characterization when there are direct sums of two or more unit-endoregular modules also unit-endoregular under certain conditions. In particular, we investigate unit-endoregular modules which are direct sums of indecomposable modules.

Clean-like properties in pullbacks and amalgamation rings

We study the transfer of the notions of clean rings, weakly clean rings, almost clean rings and unit regular rings to various context of pullbacks and amalgamations. Our attempt is to present original and new classes of these rings.

Invo-regular unital rings

It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.

Regularity and Related Rings

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.

Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings

We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. Next we study in arbitrary rings those elements whose powers are regular, and provide a method for constructing inner inverses which satisfy many additional strong relations. As a corollary we show that if each of the powers a , a 2 , … , a n a,a^2,\ldots , a^n is a regular element in some ring R R (for some n ≥ 1 n\geq 1 ), then there exists w ∈ R w\in R such that a k w k a k = a k a^k w^k a^k=a^k and w k a k w k = w k w^k a^k w^k=w^k for 1 ≤ k ≤ n 1\leq k\leq n . Similar statements are also obtained for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanness, regularity, and unit-regularity.

K 0 of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity

If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group KO(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of KO(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding KO(R) order isomorphic to G(PrimR), where PrimR is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as KO(R) for a suitable semiartinian and unit-regular ring R.

UNIT-REGULAR MODULES

AbstractIn 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

Some characterizations of ∗-regular rings

ABSTRACTA ∗-ring R is called ∗-regular if every principal one-sided ideal of R is generated by a projection. The paper is devoted to a study of ∗-regularity of ∗-rings. Basic properties of ∗-regular rings are investigated, and some equivalent characterizations on ∗-regular rings, Abelian ∗-regular rings, and *-unit regular rings are provided. We also show that a matrix ring Mn(R) is ∗-unit regular if and only if R is unit regular and is a unit for all xi in R.

On Clean Rings

A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A1 ⊕ A2 with M′ ≅ M, there is a decomposition M′ =M1 ⊕ M2 such that A = M′ ⊕ [A1 ∩ (M1 ⊕ B)] ⊕ [A2 ∩ (M2 ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.

Modules Whose Endomorphism Rings are Unit-Regular

Unit-regular rings have been extensively studied in the literature. In this article, we introduce the notion of unit endoregular modules which generalizes that of unit-regular rings. A module is called unit endoregular if its endomorphism ring is unit-regular. It is proved that a ring R is a right V-ring if and only if every finitely cogenerated right R-module is unit endoregular. Some characterizations of unit endoregular modules are provided so that several known results on unit-regular rings are generalized. We also provide some properties of unit endoregular modules. For instance, any module which is mutually subisomorphic to a unit endoregular module M is isomorphic to M.

Uniquely Clean Elements in Rings

It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules.

Inner inverses and inner annihilators in rings

For any ring element α∈R, we study the group of inner annihilators IAnn(α)={p∈R:αpα=0} and the set I(α) of inner inverses of α. For any Jacobson pair α=1−ab and β=1−ba, the groups A=IAnn(α) and B=IAnn(β) are shown to be equipotent, and A⊕C is shown to be group isomorphic to B⊕C where C=Annℓ(α)⊕Annr(α). In the case where α is (von Neumann) regular, we show further that A≅B as groups. For any Jacobson pair {α,β}, a “new Jacobson map” Φ:I(α)→I(β) is constructed that is a semigroup homomorphism with respect to the von Neumann product, and preserves units, reflexive inverses and commuting inner inverses. In particular, for any abelian ring R, Φ is a semigroup isomorphism between I(α) and I(β). As a byproduct of our methods, we also show that a ring R satisfies internal cancellation iff every Jacobson pair of regular elements are equivalent over R. In particular, the latter property holds for many rings, including semilocal rings, unit-regular rings, strongly π-regular rings, and finite von Neumann algebras.