Exchange rings, units and idempotents

Camillo, Victor; Yu, Hua-Ping

doi:10.1080/00927879408825098

Cited by 189 publications

(111 citation statements)

References 12 publications

(9 reference statements)

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“…First observe that once we show that (i) and (ii) are equivalent since Max(A) ffi Max(A Ã ) it follows that the first three are equivalent. That (ii) and (iv) are equivalent follows from the fact that bounded inversion implies the hypothesis of Proposition 10 (Camillo and Yu, 1994).…”

Section: Just For the Recordmentioning

confidence: 85%

Clean Semiprimef-Rings with Bounded Inversion

McGovern

2003

Communications in Algebra

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An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently, Anderson and Camillo (Anderson, D. D., Camillo, V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327-3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover, every clean ring is a pm-ring, that is every prime ideal is contained in a unique maximal ideal. In the same article, the authors give an example of a commutative ring which is a pm-ring yet not clean, e.g., C(R). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.

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Section: Just For the Recordmentioning

confidence: 85%

Clean Semiprimef-Rings with Bounded Inversion

McGovern

2003

Communications in Algebra

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“…We observe Following Nicholson [21], a ring is called clean if every element is a clean element (i.e., a sum of a unit and an idempotent). The class of clean rings includes semiperfect rings and unit-regular rings [4,5]. Proposition 1.5.…”

Section: Basic Structure Of Right Unit-duo Ringsmentioning

confidence: 99%

Duo Ring Property Restricted to Groups of Units

Han¹,

Lee²,

Park³

2015

Journal of the Korean Mathematical Society

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Abstract. We study the structure of right duo ring property when it is restricted within the group of units, and introduce the concept of right unit-duo. This newly introduced property is first observed to be not left-right symmetric, and we examine several conditions to ensure the symmetry. Right unit-duo rings are next proved to be Abelian, by help of which the class of noncommutative right unit-duo rings of minimal order is completely determined up to isomorphism. We also investigate some properties of right unit-duo rings which are concerned with annihilating conditions.Throughout this paper all rings are associative with identity unless otherwise stated. Let R be a ring. X(R) denotes the set of all nonzero nonunits in R, and G(R) denotes the group of all units in R., and N (R) denote the Jacobson radical, the set of all idempotents, and the set of all nilpotent elements in R, respectively. |S| denotes the cardinality of a subset S of R. Write R * = R\{0}. Z (Z n ) denotes the ring of integers (modulo n). Q denotes the field of rational numbers. GF (p n ) denotes the Galois field of order p n .Denote the n by n full (resp., upper triangular) matrix ring over R by Mat n (R) (resp., U n (R)) and use e ij for the matrix with (i, j)-entry 1 and elsewhere 0. Following the literature, we write D n (R) = {(a ij ) ∈ U n (R) | all diagonal entries are equal } and V n (R) = {(a ij ) ∈ D n (R) | a 1k = a 2(k+1) = · · · = a hn for h = 1, 2, . . . , n − 1 and k = 2, . . . , n}.Due to Feller [7], a ring is called right (resp. left) duo if every right (resp. left) ideal is an ideal; a ring is called duo if it is both right and left duo. It is easily shown that idempotents of right duo rings are central. There are very useful results for duo rings in [3,16,24].

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“…(1) Let R = {(q 1 , q 2 , ..., q n , z, z, z, ...) : n ≥ 1, q i ∈ Q and z ∈ Z 2 } where Z 2 denotes the localization of Z at the prime ideal (2). Then R is a VNL ring.…”

Section: Examples 21mentioning

confidence: 99%

“…In Section 4, we characterize arbitrary VNL rings which do not have infinite set of orthogonal idempotents. As an exchange ring without infinite set of orthogonal idempotents is semiperfect (see [2]), this gives us characterization of semiperfect VNL rings. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a 1 , a 2 ) ∈ R 2 , one of the a i is regular in R. We also characterize VNL rings R with a primitive idempotent e such that eRe is not a division ring or equivalently J(eRe) = 0 (if e is a primitive idempotent in an exchange ring R, then eRe is a local ring).…”

Section: Introductionmentioning

confidence: 99%

Some Characterizations of VNL Rings

Grover

Khurana

2009

Communications in Algebra

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A ring R is said to be VNL if for any a ∈ R, either a or 1−a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without infinite set of orthogonal idempotents; and also the VNL rings having primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a 1 , a 2 ) ∈ R 2 , one of the a i is regular in R. Formal triangular matrix rings that are VNL, are also characterized. As a corollary it is shown that an upper triangular matrix ring T n (R) is VNL if and only if n = 2 or 3 and R is a division ring.

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Exchange rings, units and idempotents

Cited by 189 publications

References 12 publications

Clean Semiprimef-Rings with Bounded Inversion

Clean Semiprimef-Rings with Bounded Inversion

Duo Ring Property Restricted to Groups of Units

Some Characterizations of VNL Rings

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