Extensions of Exchange Rings

Ara, Pere

doi:10.1006/jabr.1997.7116

Cited by 97 publications

(87 citation statements)

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“…[1][2]), an ideal I of a ring R is an exchange ideal provided that for every x ∈ I there exist an idempotent e ∈ I and elements r, s ∈ I such that e = xr = x + s − xs. Let I be an exchange ideal of a ring R, and let e ∈ R be an idempotent.…”

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confidence: 99%

On Separative Refinement Monoids

Chen¹

2009

Bulletin of the Korean Mathematical Society

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Abstract. We obtain two new characterizations of separativity of refinement monoids, in terms of comparability-type conditions. As applications, we get several equivalent conditions of separativity for exchange ideals.A commutative monoid (M, +, 0) has a refinement (or is a refinement monoid) if, for all a, b, c and d in M , the equation a+b = c+d implies the existence ofThese equations are represented in the form of a refinement matrix: Refinement monoids have been extensively studied in recent years (cf. [4] and [7]). A commutative monoidSeparativity is a weak form of cancellativity for commutative monoids. Many authors have studied separative refinement monoids from various view-points (see [3][4] and [6][7]). In this article, we get two new characterizations of separative refinement monoids. We prove that every separative refinement monoid can be characterized by a certain sort of comparability. Also we introduce the concept of refinement extensions of a refinement monoid. We see that every separative refinement monoid can be characterized by such refinement extensions. Let I be an ideal of a ring R. We use F P (I) to denote the class of finitely generated projective right R-modules P with P = P I and V (I) to denote the monoid of isomorphism classes of objects from F P (I). Following Ara et al. (see [3]), an exchange ideal I of a ring R is separative if V (I) is a separative refinement monoid, that is, for any A, B, C ∈ F P (I),We say that R is a separative ring if R is separative as an ideal of R.Separativity plays a key role in the direct sum decomposition theory of exchange rings. It seems rather likely that non-separative exchange rings should exist. We say that an exchange ring R satisfies the comparability axiom provided that, for any finitely generated projective right R-modules A and B, either A ⊕ B or B ⊕ A. In [7, Theorem 3.9], Pardo showed that every

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confidence: 99%

On Separative Refinement Monoids

Chen¹

2009

Bulletin of the Korean Mathematical Society

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“…According to Ara [1], a ring I (without unit) is called an exchange ring if for each a ∈ I there exist an idempotent e ∈ I and r, s ∈ I such that e = ar = a + s − as. Also if I is an ideal of a unital exchange ring, then I satisfies the above condition.…”

Section: Corollary 310 Let R Be An Abelian Ring With Only a Finite mentioning

confidence: 99%

On Noncommutative VNL-Rings and GVNL-Rings

Chen¹,

Tong²

2006

Glasgow Math. J.

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Abstract. It is proved that every abelian VNL-ring is an SVNL-ring, which gives a positive answer to a question of Osba et al. [7]. Some characterizations of duo VNLrings are given and some main results of Osba et al. [7] on commutative VNL-rings are extended to right duo VNL-rings and even abelian GVNL-rings.2000 Mathematics Subject Classification. 16D25, 16E50. Introduction.Rings considered are associative with identity unless the contrary is stated explicitly. An element a of a ring R is π -regular if there exist a positive integer n and x ∈ R such that a n = a n xa n . In the case of n = 1, a is regular. A ring R is π -regular (regular) if every element in R is π -regular (regular). A ring R is an exchange ring if for every a ∈ R there exists an idempotent e ∈ R such that e ∈ aR and (1 − e) ∈ (1 − a)R. A ring R is right (left) duo if every right (left) ideal is two-sided, and R is duo if R is right and left duo. A ring R is abelian if all idempotents are contained in the center and a ring R is reduced if it does not contain nonzero nilpotent elements.Following Contessa [4], a commutative ring R is a VNL-ring if for every a ∈ R, at least one of a or 1 − a is regular. According to Osba et al. [7], a commutative ring R is an SVNL-ring if whenever (S) = R for some nonempty subset S of R, at least one of the elements in S is regular, where (S) is an ideal generated by S. Some properties of VNL-rings and SVNL-rings are investigated in Osba et al. [7]. But they are unable to solve the question whether every VNL-ring is an SVNL-ring. Because of this they are unable to characterize VNL-rings abstractly in the sense of relating them to more familiar classes of rings. In the present paper we define a noncommutative ring R to be a VNL-ring (GVNL-ring) if for every a ∈ R at least one of a or 1 − a is regular (π -regular) and a ring R is an SVNL-ring, if whenever (S) r = R for some nonempty subset S of R, at least one element in S is regular, where (S) r is a right ideal generated by S. The main purpose of this paper is to prove that every abelian VNL-ring is an SVNL-ring, giving an answer to the question of Osba et al. [7] in the affirmative. Furthermore we give some characterizations of duo VNL-rings, and extend and improve some main results of Osba et al. [7] on commutative VNL-rings.Throughout this paper we use the symbol J(R) to denote the Jacobson radical of a ring R, Id(R) its set of idempotents, and Max(R) its maximal spectrum. The left annihilator of an element a in R is denoted by A l (a). For an integer n ≥ 2, the symbol

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“…Thus we say that a (not necessarily unital) ring R is an exchange ring if for each x in R there is an idempotent e in xR such that x = e + y − ey for some y in R. This idea was successfully exploited by Ara in [1]. If I is a non-unital exchange ring embedded as a two-sided ideal of a unital ring R, we will adopt the terminology from [1] and say that I is an exchange ideal of R. Lemma 1.2. Let I be a two-sided exchange ideal in a unital ring R. If x and y are elements in R and p is an idempotent such that p − xy ∈ I, there is an idempotent q in I and an element r in pRp with p − r in I, such that both elements a = (p − q)x and b = yr are regular and partial inverses for one another.…”

Section: Exchange Ringsmentioning

confidence: 99%