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Displaying similar documents to “A canonical directly infinite ring”

When is every order ideal a ring ideal?

Melvin Henriksen, Suzanne Larson, Frank A. Smith (1991)

Commentationes Mathematicae Universitatis Carolinae

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A lattice-ordered ring ℝ is called an if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those f -rings ℝ such that ℝ / 𝕀 is contained in an f -ring with an identity element that is a strong order unit for some nil l -ideal 𝕀 of ℝ . In particular, if P ( ℝ ) denotes the set of nilpotent elements of the f -ring ℝ , then ℝ is an OIRI-ring if and only if ℝ / P ( ℝ ) is contained in an f -ring with an identity element that is a strong order unit. ...

Conditions under which R ( x ) and R 〈 x 〉 are almost Q-rings

Hani A. Khashan, H. Al-Ezeh (2007)

Archivum Mathematicum

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All rings considered in this paper are assumed to be commutative with identities. A ring R is a Q -ring if every ideal of R is a finite product of primary ideals. An almost Q -ring is a ring whose localization at every prime ideal is a Q -ring. In this paper, we first prove that the statements, R is an almost Z P I -ring and R [ x ] is an almost Q -ring are equivalent for any ring R . Then we prove that under the condition that every prime ideal of R ( x ) is an extension of a prime ideal of R , the ring R ...

Diagonal reductions of matrices over exchange ideals

Huanyin Chen (2006)

Czechoslovak Mathematical Journal

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In this paper, we introduce related comparability for exchange ideals. Let I be an exchange ideal of a ring R . If I satisfies related comparability, then for any regular matrix A ∈ M n ( I ) , there exist left invertible U 1 , U 2 ∈ M n ( R ) and right invertible V 1 , V 2 ∈ M n ( R ) such that U 1 V 1 A U 2 V 2 = diag ( e 1 , ⋯ , e n ) for idempotents e 1 , ⋯ , e n ∈ I .

Classes of Commutative Clean Rings

Wolf Iberkleid, Warren Wm. McGovern (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let A be a commutative ring with identity and I an ideal of A . A is said to be I - c l e a n if for every element a ∈ A there is an idempotent e = e 2 ∈ A such that a - e is a unit and a e belongs to I . A filter of ideals, say ℱ , of A is if for each I ∈ ℱ there is a finitely generated ideal J ∈ ℱ such that J ⊆ I . We characterize I -clean rings for the ideals 0 , n ( A ) , J ( A ) , and A , in terms of the frame of multiplicative Noetherian filters of ideals of A , as well as in terms of more classical ring properties.