A GENERALIZATION OF REDUCED RINGS

For a ring derivation δ, we introduce and investigate a generalization of reduced rings and Armendariz rings which we call a δ-Armendariz ring. Various classes of δ-Armendariz rings is provided and a number of properties of this generalization are established. Radicals and minimal prime ideals of the differential polynomial ring R[x; δ], in terms of those of a δ-Armendariz R, is determined. We prove that several properties transfer between R and the differential polynomial ring R[x; δ], in case R is δ-Armendariz.

[1]  Yang Lee,et al.  Radicals of skew polynomial rings and skew Laurent polynomial rings , 2011 .

[2]  N. Mahdou,et al.  On Armendariz rings. , 2009 .

[3]  A. Moussavi,et al.  Baer and Quasi-Baer Differential Polynomial Rings , 2008 .

[4]  Yiqiang Zhou,et al.  Armendariz and Reduced Rings , 2004 .

[5]  Ken R. Goodearl,et al.  An Introduction to Noncommutative Noetherian Rings: Prologue , 2004 .

[6]  G. Marks.  A taxonomy of 2-primal rings , 2003 .

[7]  Y. Hirano.  On annihilator ideals of a polynomial ring over a noncommutative ring , 2002 .

[8]  A. Smoktunowicz,et al.  ARMENDARIZ RINGS AND SEMICOMMUTATIVE RINGS , 2002 .

[9]  G. Birkenmeier,et al.  Polynomial extensions of Baer and quasi-Baer rings , 2001 .

[10]  G. Marks.  ON 2-PRIMAL ORE EXTENSIONS , 2001 .

[11]  J. McConnell,et al.  Noncommutative Noetherian Rings , 2001 .

[12]  G. Birkenmeier,et al.  PRINCIPALLY QUASI-BAER RINGS , 2001 .

[13]  A. Smoktunowicz.  Polynomial Rings over Nil Rings Need Not Be Nil , 2000 .

[14]  Yang Lee,et al.  Armendariz Rings and Reduced Rings , 2000 .

[15]  Y. Hirano,et al.  Semiprime ore extensions , 2000 .

[16]  D. D. Anderson,et al.  Semigroups and rings whose zero products commute , 1999 .

[17]  G. Marks.  Skew polynomial rings over 2-primal rings , 1999 .

[18]  D. D. Anderson,et al.  Armendariz rings and gaussian rings , 1998 .

[19]  Yang Lee,et al.  Questions on 2-primal rings , 1998 .

[20]  T. Lam,et al.  Primeness, semiprimeness and prime radical of ore extensions , 1997 .

[21]  G. Marks.  Direct Product and Power Series Formations Over 2-Primal Rings , 1997 .

[22]  Tsit Yuen Lam,et al.  A first course in noncommutative rings , 2002 .

[23]  C. Ferran.  Zip rings and mal'cev domains , 1991 .

[24]  J. W. Kerr.  The polynomial ring over a Goldie ring need not be a Goldie ring , 1990 .

[25]  C. Faith.  Rings with zero intersection property on annihilators : zip rings. , 1989 .

[26]  H. Koo,et al.  Isomorphic ore extensions , 1987 .

[27]  K. Kishimoto,et al.  On differential rings and skew pulynomials , 1985 .

[28]  K. Kishimoto,et al.  On Radicals of Skew Polynomial Rings of Derivation Type , 1983 .

[29]  J. Zelmanowitz.  The finite intersection property on annihilator right ideals , 1976 .

[30]  D. Jordan.  Noetherian Ore Extensions and Jacobson Rings , 1975 .

[31]  E. Armendariz.  A note on extensions of Baer and P. P. -rings , 1974, Journal of the Australian Mathematical Society.

[32]  Gooyong Shin.  Prime ideals and sheaf representation of a pseudo symmetric ring , 1973 .

[33]  H. Bell.  Near-rings in which each element is a power of itself , 1970, Bulletin of the Australian Mathematical Society.

[34]  Abraham Zaks,et al.  On Baer and quasi-Baer rings , 1970 .

[35]  W. Clark.  Twisted matrix units semigroup algebras , 1967 .

[36]  S. A. Amitsur.  Radicals Of Polynomial Rings , 1956, Canadian Journal of Mathematics.

[37]  I. Kaplansky.  Rings of operators , 1956 .

[38]  A. S. Amitsur.  Algebras over infinite fields , 1956 .