Trivial extensions subject to semi-regularity and semi-coherence

Abstract In this paper, we investigate the transfer of Matlis’ semi-regularity and semi-coherence in trivial ring extensions issued from rings (with zero-divisors). We use the obtained results to enrich the literature with new examples of semi-regular or semi-coherent rings issued as trivial extensions and validate some questions left open in the literature.

[1]  M. Tamekkante,et al.  Note on the Weak Global Dimension of Coherent Bi-amalgamations , 2017 .

[2]  S. Kabbaj,et al.  Zaks' conjecture on rings with semi-regular proper homomorphic images , 2016, 1604.03268.

[3]  S. Kabbaj,et al.  Matlis' semi-regularity in trivial ring extensions issued from integral domains , 2016, 1604.02795.

[4]  N. Mahdou,et al.  Trivial Ring Extensions Defined by Arithmetical-Like Properties , 2013 .

[5]  B. Olberding.  Prescribed subintegral extensions of local Noetherian domains , 2013, 1306.0870.

[6]  N. Mahdou,et al.  SELF INJECTIVE AMALGAMATED DUPLICATION OF A RING ALONG AN IDEAL , 2013 .

[7]  B. Olberding.  A counterpart to Nagata idealization , 2012, 1204.3962.

[8]  S. Goto,et al.  Almost Gorenstein rings , 2011, 1106.1301.

[9]  N. Mahdou,et al.  IF-dimension of Modules , 2010, Communications in Mathematics and Applications.

[10]  D. D. Anderson,et al.  Idealization of a Module , 2009 .

[11]  M. Fontana,et al.  Transfinite self-idealization and commutative rings of triangular matrices , 2009 .

[12]  Farid Kourki.  Sur les Extensions Triviales Commutatives , 2009 .

[13]  S. Kabbaj,et al.  Commutative rings in which every finitely generated ideal is quasi-projective , 2008, 0810.0359.

[14]  N. Mahdou,et al.  Trivial extensions defined by Prüfer conditions , 2008, 0808.0275.

[15]  Sarah Glaz,et al.  Gaussian properties of total rings of quotients , 2007 .

[16]  J. Brewer,et al.  Multiplicative ideal theory in commutative algebra : a tribute to the work of Robert Gilmer , 2006 .

[17]  N. Mahdou,et al.  Trivial Extensions Defined by Coherent-like Conditions , 2004, math/0606696.

[18]  F. Couchot.  Injective modules and fp-injective modules over valuation rings , 2003, math/0409519.

[19]  R. Y. Sharp,et al.  MODULES OVER NON-NOETHERIAN DOMAINS (Mathematical Surveys and Monographs 84) By LÁSZLÓ FUCHS and LUIGI SALCE: 613 pp., US$109.00, ISBN 0-8218-1963-1 (American Mathematical Society, Providence, RI, 2001) , 2002 .

[20]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[21]  Sarah Glaz.  Finite conductor rings , 2000 .

[22]  L. Fuchs,et al.  Modules over non-Noetherian domains , 2000 .

[23]  Sarah Glaz,et al.  Commutative Coherent Rings , 1989 .

[24]  J. Huckaba.  Commutative Rings with Zero Divisors , 1988 .

[25]  Dorin Popescu,et al.  General Néron desingularization , 1985, Nagoya Mathematical Journal.

[26]  G. Levin.  Modules and Golod homomorphisms , 1985 .

[27]  Eben Matlis.  Commutative semi-coherent and semi-regular rings , 1985 .

[28]  R. Damiano,et al.  Commutative torsion stable rings , 1984 .

[29]  M. Harada.  Self mini-injective rings , 1982 .

[30]  S. Goto.  Approximately Cohen-Macaulay Rings , 1982 .

[31]  J. Roos.  Finiteness conditions in commutative algebra and solution of a problem of Vasconcelos. , 1982 .

[32]  R. Colby.  Rings which have flat injective modules , 1975 .

[33]  Tor H. Gulliksen,et al.  A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity. , 1974 .

[34]  J. Roos,et al.  Explicit formulae for the global homological dimensions of trivial extensions of rings , 1973 .

[35]  Saroj Jain.  Flat and FP-injectivity , 1973 .

[36]  R. Fossum.  Commutative extensions by canonical modules are Gorenstein rings , 1973 .

[37]  I. Reiten.  The converse to a theorem of Sharp on Gorenstein modules , 1972 .

[38]  Gabriel Sabbagh,et al.  Embedding Problems for Modules and Rings with Application to Model-Companions , 1971 .

[39]  C. U. Jensen.  Arithmetical rings , 1966 .

[40]  Hubert S. Butts,et al.  Prüfer rings , 1966 .

[41]  L. Fuchs.  Über die Ideale arithmetischer Ringe , 1949 .

[42]  Masayoshi Nagata,et al.  Local Rings , 2022 .