On commutative P. P. rings

Evans, Michael

doi:10.2140/pjm.1972.41.687

Cited by 60 publications

(38 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, from the result, it is easily seen that every central idempotent of S must be an idempotent ofR. This result therefore generalizes the result of Bergman (1971) and Evans (1972).…”

Section: For a Ring R The Following Are Equivalent: (I) R Is A Rightsupporting

confidence: 72%

Extended semi-hereditary rings

Evans¹

1978

J. Aust. Math. Soc.

View full text Add to dashboard Cite

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

show abstract

“…However, from the result, it is easily seen that every central idempotent of S must be an idempotent ofR. This result therefore generalizes the result of Bergman (1971) and Evans (1972).…”

Section: For a Ring R The Following Are Equivalent: (I) R Is A Rightsupporting

confidence: 72%

Extended semi-hereditary rings

Evans¹

1978

J. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…097 ( ± 0°. 001), which was ∼74°from the LAT boresight at the burst trigger time (Evans & Osborne 2013). We show the GBM light curves of GRB 131014A in several energy ranges in Figure 1.…”

Section: Observationsmentioning

confidence: 99%

GRB 131014A: A LABORATORY FOR STUDYING THE THERMAL-LIKE AND NON-THERMAL EMISSIONS IN GAMMA-RAY BURSTS, AND THE NEW ${L}_{{\rm{i}}}^{\mathrm{nTh}}-{E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ RELATION

Guiriec

Mochkovitch

Piran

et al. 2015

ApJ

View full text Add to dashboard Cite

Over the past years, evidence has been accumulated on the existence of a thermal-like component during the prompt phase of Gamma Ray Bursts (GRBs). However, this component, often associated with the GRB jet's photosphere, is usually subdominant compared to a much stronger non-thermal one. The prompt emission of GRB 131014A-detected by the Fermi Gamma-ray Space Telescope (hereafter Fermi)-provides an unique opportunity to trace the history of this thermal-like component. Indeed, the thermal emission in GRB 131014A is much more intense than in other GRBs and a pure thermal episode is observed during the initial 0.16 s. The thermal-like component cools monotonically during the first second while the non-thermal emission kicks off. The intensity of the non-thermal component progressively increases until being energetically dominant at late time similarly to what is typically observed. This is a perfect scenario to disentangle the thermal component from the nonthermal one. The initial decaying and cooling phase of the thermal-like component is followed by a strong re-brightening and a re-heating episode; however, despite a much brighter second emission phase, the temperature of the thermal component does not reach its initial value. This re-brightening episode is followed by a global constant cooling until the end of the burst. We note that a shallower low-energy spectral slope than the typical index value +1, corresponding to a pure Planck function, better matches with the thermal-like spectral shape; a spectral index around +0.6 seems to be in better agreement with the data. The non-thermal component is adequately fitted with a Band function whose low and high energy power law indices are ∼-0.7 and <∼-3, respectively; this is also statistically globally equivalent to a cutoff power law with a ∼-0.7 index. This is in agreement with our previous results. Finally, a strong correlation is observed between the time-resolved energy flux, F nTh i , and the corresponding spectral peak energy, E nTh peak,i , of the non-thermal component with a slope similar to the one reported in our previous articles. Assuming a universal relation between the time-resolved luminosity of the non-thermal component, L nTh i , and its rest frame E nTh peak,i , E rest,nTh peak,i , that we derived from a limited sample of GRBs detected by Fermi, we estimate a redshift of ∼1.55 for GRB 131014A that is a typical value for long GRBs. These observational results are consistent with the models in which the non-thermal emission is produced well above the GRB jet photosphere but they may also be compatible with other scenarios (e.g., dissipative photosphere) that are not discussed in this article.

show abstract

“…if every principal ideal of R is projective (see [4]). (ii) A ring R is said to be semihereditary if every finitely generated ideal of R is projective.…”

Section: A Characterization Of Prüfer Ringsmentioning

confidence: 99%

A criterion for rings which are locally valuation rings

Divaani-Aazar

Tousi

2009

Colloq. Math.

View full text Add to dashboard Cite

Abstract. Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically pure injective.

show abstract

On commutative P. P. rings

Cited by 60 publications

References 16 publications

Extended semi-hereditary rings

Extended semi-hereditary rings

GRB 131014A: A LABORATORY FOR STUDYING THE THERMAL-LIKE AND NON-THERMAL EMISSIONS IN GAMMA-RAY BURSTS, AND THE NEW ${L}_{{\rm{i}}}^{\mathrm{nTh}}-{E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ RELATION

A criterion for rings which are locally valuation rings

Contact Info

Product

Resources

About