Hereditary Rings

Small, Lance W.

doi:10.1073/pnas.55.1.25

Cited by 27 publications

(8 citation statements)

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“…The equivalence of (iii) and (iv) is known (Stenstrom (1975) PROOF. This follows immediately from Small (1967) and Theorem 3.8 of Stone (1970), which establishes that if Q n is right torsion-free as an /{ n -module for all n> 1, then Q is right flat as an /{-module.…”

Section: Extended Semi-hereditary Ringsmentioning

confidence: 74%

Extended semi-hereditary rings

Evans¹

1978

J. Aust. Math. Soc.

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A ring R for which every finitely generated right submodule of S R , the left flat epimorphic hull of R, is projective is termed an extended semi-hereditary ring. It is shown that several of the characterizing properties of Prufer domains may be generalized to give characterizations of extended semi-hereditary rings. A suitable class of PP rings is introduced which in this case serves as a generalization of commutative integral domains. A ring will be said to be a right perfect PP ring if S, the left flat epimorphic hull of R is a regular ring and every cyclic i?-submodule of S R is projective. A ring R will be said to be a right extended semi-hereditary ring if S, the left flat epimorphic hull of R, is regular and every finitely generated right jR-submodule of S R is projective.Clearly every right extended semi-hereditary ring is a right perfect PP ring. In this paper characterizations of both of these classes of rings are given and the ideal structure is discussed. In Section 3 characterizations are given for right extended semi-hereditary rings which are analogous to the "classical" characterizations of Prufer domains.In Hattori (1960) a right .R-module A B was defined to be torsion-free if for all aeA and xeR, ax = 0 implies aeAl^x). Throughout this paper torsion-free will 465 use, available at https://www.cambridge.org/core/terms. https://doi

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Section: Extended Semi-hereditary Ringsmentioning

confidence: 74%

Extended semi-hereditary rings

Evans¹

1978

J. Aust. Math. Soc.

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“…Chase [12] in 1961. According to Chase, a semiprimary ring A with Jacobson radical R is triangular if there exists a complete set e 1 ; e 2 ; : : : ; e k of mutually orthogonal primitive idempotents of A such that e i Ae j D 0 for all i > j: This notion was used by L. W. Small in [38] for arbitrary right Noetherian rings. In 1966, M. Harada [23] introduced the term "generalized triangular matrix rings" for rings with triangular decomposition of the identity, where A i D e i Ae i are arbitrary rings.…”

Section: Remark 42mentioning

confidence: 99%

Rings with finite decomposition of identity

Dokuchaev¹,

Gubareni²,

Kirichenko³

2011

Ukr Math J

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A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also present a brief survey of some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is distributive. An application of decompositions of identity to groups of units is given.

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“…Proof. We have R/P is a right hereditary prime ring, and by Theorem 3 of [12], then R/P is a hereditary Noetherian prime ring. Consequently, by 3.52 of [6, p. 310], then \R/P\ = 0 or 1.…”

Section: Let R Be An A-coprimitive Ring With Unique A-prime P Then Pmentioning

confidence: 99%