Von Neumann Regular Rings and Direct Sum Decomposition Problems

Goodearl, K. R.

doi:10.1007/978-94-011-0443-2_20

Cited by 79 publications

(109 citation statements)

References 17 publications

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“…Also the identity in 7\ is a sum of two units in Ti. D We know that a right self-injective ring is right continuous (see [4]). The following example shows that Lemma 5 is not true even for commutative regular continuous rings.…”

Section: Is Not Injective Then V Extends To An Automorphism V' Of Anmentioning

confidence: 99%

“…Then R is a commutative continuous ring (see [4,Example 13.8]) which clearly is regular. It is easy to see that the identity of R is not a sum of two units.…”

mentioning

confidence: 99%

“…An element (x n ) N of R available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700039289 [4] is an idempotent precisely when every component is either 0 or 1 and so it is clear that eR is not a Boolean ring for any idempotent e € R. Also the element (1,0,1,0,...) is not a sum of units in R implying that usn(J2) = oo. The following result characterises the regular right self-injective rings with various unit sum numbers.…”

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confidence: 99%

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Unit sum numbers of right self-injective rings

Khurana¹,

Srivastava²

2007

Bull. Austral. Math. Soc.

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In a recent paper (which is to appear in J. Algebra Appl.) we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to Z 2 and hence the unit sum number of a nonzero right self-injective ring is 2, w or oo. In this paper we characterise right self-injective rings with unit sum numbers w and oo. We prove that the unit sum number of a right self-injective ring R is w if and only if R has a factor ring isomorphic to Z 2 but no factor ring isomorphic to Z2 x Z 2 , and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is 00 precisely when R has a factor ring isomorphic to Z2 x Z 2 . We also answer a question of Henriksen (which appeared in J. Algebra, Question E, page 192), by giving a large class of regular right self-injective rings having the unit sum number w in which not all non-invertible elements are the sum of two units.We shall consider associative rings with identity. Our modules will be unital right modules with endomorphisms acting on the left.A ring R is said to have the n-sum property, for a positive integer n, if each of its elements can be written as a sum of exactly n units of R. It is obvious that a ring having the n-sum property also has the fc-sum property for every positive integer k > n. The unit sum number of a ring R, denoted by usn(/J), is the least integer n, if it exists, such that R has the n-sum property. If R has an element which is not a sum of units then we set usn(i?) to be 00, and if every element of R is a sum of units but R does not have the n-sum property for any n, then we set usn(/?) = u. Clearly, usn(i?) = 1 if and only if R has only one element. The unit sum number of a module M, denoted by usn(M), is the unit sum number of its endomorphism ring. The topic has been studied extensively (see [2,3,5,6,7,10,11,15,17,19,20]).

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Section: Is Not Injective Then V Extends To An Automorphism V' Of Anmentioning

confidence: 99%

“…Then R is a commutative continuous ring (see [4,Example 13.8]) which clearly is regular. It is easy to see that the identity of R is not a sum of two units.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Unit sum numbers of right self-injective rings

Khurana¹,

Srivastava²

2007

Bull. Austral. Math. Soc.

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“…Moreover, if R is regular, then V(R) satisfies the refinement property. Furthermore, we equip V(R) with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y. References about this can be found in [1], [5], [6], [10]. Proof.…”

Section: Applications To Von Neumann Regular Ringsmentioning

confidence: 99%

A uniform refinement property for congruence lattices

Wehrung¹

1999

Proc. Amer. Math. Soc.

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Abstract. The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt's construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tůma. In a previous paper, we constructed a distributive algebraic lattice A with ℵ 2 compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice A cannot be obtained using the Pudlák, Tischendorf, Tůma approach.The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.

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“…is still open, despite a strong consensus that non-separative exchange rings should exist. See [3,4,7] for more background on this problem.…”

Section: Introductionmentioning

confidence: 99%

Gromov translation algebras over discrete trees are exchange rings

Ara¹,

O’Meara²,

Perera³

2003

Trans. Amer. Math. Soc.

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Abstract. It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras G(0) of ω × ω matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers r in the unit interval [0, 1], the growth algebras G(r) (introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension r in [0,1].

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Von Neumann Regular Rings and Direct Sum Decomposition Problems

Cited by 79 publications

References 17 publications

Unit sum numbers of right self-injective rings

Unit sum numbers of right self-injective rings

A uniform refinement property for congruence lattices

Gromov translation algebras over discrete trees are exchange rings

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