Ore Extension Rings with Constant Products of Elements
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Let R be an associative unital ring with an endomorphism α and α-derivation δ . The constant products of elements in Ore extension rings, when the coe cient ring is reversible, is investigated. We show that if f(x) = Σ^n_i=0 a_ix^i and g(x) =Σ^m_j=0 b_jx^j be non-zero elements in Ore extension ring R[x;α;δ] such that g(x)f(x) = c∈R, then there exist non-zero elements r, a∈R such that rf(x) = ac, when R is an (α,δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x;α;δ], when the coe cient ring R is (α,δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α,δ)-compatible, then J(R[x;α;δ]) = Nil(R)[x;α;δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the nal part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.
Abstract
Let R be an associative unital ring with an endomorphism α and α-derivation δ . The constant products of elements in Ore extension rings, when the coe cient ring is reversible, is investigated. We show that if f(x) = Σ^n_i=0 a_ix^i and g(x) =Σ^m_j=0 b_jx^j be non-zero elements in Ore extension ring R[x;α;δ] such that g(x)f(x) = c∈R, then there exist non-zero elements r, a∈R such that rf(x) = ac, when R is an (α,δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x;α;δ], when the coe cient ring R is (α,δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α,δ)-compatible, then J(R[x;α;δ]) = Nil(R)[x;α;δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the nal part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.
참고문헌 (18)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc, 2, pp.363~368 (1970)
- G. F. Birkenmeier, "Completely prime ideals and associated radicals, Ring Theory", pp.102~129 (1993), World Sci. Publ
- W. X. Chen, "On constant products of elements in skew polynomial rings", Bull. Iranian Math. Soc, 41(2), pp.453~462 (2015)
- W. X. Chen, "On weakly semicommutative rings", Commun. Math. Res, 27(2), pp.179~192 (2011)
- J. Chen, "On strongly clean matrix and triangular matrix rings", Comm. Algebra, 34(10), pp.3659~3674 (2006)
- M. Habibi, "The McCoy condition on Ore extensions", Comm. Algebra, 41(1), pp.124~141 (2013)
- E. Hashemi, "Compatible ideals and radicals of Ore extensions", New York J. Math, 12, pp.349~356 (2006)
- E. Hashemi, "Some types of ring elements in Ore extensions over noncommutative rings", J. Algebra Appl, 16(11), pp.1750201~ (2017)[DOI]
- E. Hashemi, "Polynomial extensions of quasi-Baer rings", Acta Math. Hungar, 107(3), pp.207~224 (2005)
- Y. Hirano, "Some studies on strongly π-regular rings", Math. J. Okayama Univ, 20(2), pp.141~149 (1978)
- C. Y. Hong, "Ore extensions of Baer and p. p. -rings", J. Pure Appl. Algebra, 151(3), pp.215~226 (2000)
- C. Y. Hong, "Rigid ideals and radicals of Ore extensions", Algebra Colloq, 12(3), pp.399~412 (2005)
- O. A. S. Karamzadeh, "On constant products of polynomials", Int. J. Math. Edu, 18, pp.627~629 (1987)
- J. Krempa, "Some examples of reduced rings", Algebra Colloq, 3(4), pp.289~300 (1996)
- N. H. McCoy, "Remarks on divisors of zero", Amer. Math. Monthly, 49, pp.286~295 (1942)
- A. R. Nasr-Isfahani, "Ore extensions of 2-primal rings", J. Algebra Appl, 13(3), pp.1350117~ (2014)[DOI]
- P. P. Nielsen, "Semi-commutativity and the McCoy condition", Algebra, 298(1), pp.134~141 (2006)
- G. Y. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Trans. Amer. Math. Soc, 184, pp.43~60 (1973)