ON QUASI-RIGID IDEALS AND RINGS
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Let σ be an endomorphism and I a σ-ideal of a ring R. Pearson and Stephenson called I a σ-semiprime ideal if whenever A is an ideal of R and m is an integer such that Aσt(A) ⊆ I for all t ≥ m, then A ⊆ I,where σ is an automorphism, and Hong et al. called I a σ-rigid ideal if aσ(a) ∈ I implies a ∈ I for a ∈ R. Notice that R is called a σ-semiprime ring (resp., a σ-rigid ring) if the zero ideal of R is a σ-semiprime ideal (resp., a σ-rigid ideal). Every σ-rigid ideal is a σ-semiprime ideal for an automorphism σ, but the converse does not hold, in general. We, in this paper, introduce the quasi σ-rigidness of ideals and rings for an automorphism σ which is in between the σ-rigidness and the σ-semiprimeness,and study their related properties. A number of connections between the quasi σ-rigidness of a ring R and one of the Ore extension R[x; σ, δ] of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if R[x; σ, δ] is a (principally) quasi-Baer ring, when R is a quasi σ-rigid ring.
Abstract
Let σ be an endomorphism and I a σ-ideal of a ring R. Pearson and Stephenson called I a σ-semiprime ideal if whenever A is an ideal of R and m is an integer such that Aσt(A) ⊆ I for all t ≥ m, then A ⊆ I,where σ is an automorphism, and Hong et al. called I a σ-rigid ideal if aσ(a) ∈ I implies a ∈ I for a ∈ R. Notice that R is called a σ-semiprime ring (resp., a σ-rigid ring) if the zero ideal of R is a σ-semiprime ideal (resp., a σ-rigid ideal). Every σ-rigid ideal is a σ-semiprime ideal for an automorphism σ, but the converse does not hold, in general. We, in this paper, introduce the quasi σ-rigidness of ideals and rings for an automorphism σ which is in between the σ-rigidness and the σ-semiprimeness,and study their related properties. A number of connections between the quasi σ-rigidness of a ring R and one of the Ore extension R[x; σ, δ] of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if R[x; σ, δ] is a (principally) quasi-Baer ring, when R is a quasi σ-rigid ring.
참고문헌 (22)
- S. Annin, "Associated primes over skew polynomial rings", Comm. Algebra, 30(5), pp.2511~2528 (2002)
- Muhittin Baser, "Generalized semicommutative rings and their extensions", 대한수학회보, 45(2), pp.285~297 (2008), 대한수학회
- G. F. Birkenmeier, "Completely prime ideals and associated radicals. in: Ring theory", pp.102~129 (1993), World Sci. Publ.
- G. F. Birkenmeier, "Principally quasi-Baer rings", Comm. Algebra, 29(2), pp.636~660 (2001)
- G. F. Birkenmeier, "Polynomial extensions of Baer and quasi-Baer rings", J. Pure Appl. Algebra, 159(1), pp.25~42 (2001)
- W. E. Clark, "Twisted matrix units semigroup algebras", Duke Math. J., 34, pp.417~423 (1969)
- E. Hashemi, "Compatible ideals and radicals of Ore extensions", New York J. Math., 12, pp.349~356 (2006)
- E. Hashemi, "Polynomial extensions of quasi-Baer rings", Acta Math. Hungar., 107(3), pp.207~224 (2005)
- C. Y. Hong, "Ore extensions of Baer and p.p.-rings", J. Pure Appl. Algebra, 151(3), pp.215~226 (2000)
- C. Y. Hong, "On skew Armendariz rings", Comm. Algebra, 31(1), pp.103~122 (2003)
- C. Y. Hong, "Ore extensions of quasi-Baer rings", Comm. Algebra, 37, pp.2030~2039 (2009)
- C. Y. Hong, "On minimal strongly prime ideals", Comm. Algebra, 28(10), pp.4867~4878 (2000)
- C. Y. Hong, "Rigid ideals and radicals of Ore extensions", Algebra Colloq., 12(3), pp.399~412 (2005)
- A. A. M. Kamal, "Some remarks on Ore extension rings", Comm. Algebra, 22(10), pp.3637~3667 (1994)
- I. Kaplansky, "Rings of operators" (1968), W. A. Benjamin, Inc.
- N. K. Kim, "Minimal prime ideals in 2-primal rings", Math. Japon., 50(3), pp.415~420 (1999)
- J. Krempa, "Some examples of reduced rings", Algebra Colloq., 3(4), pp.289~300 (1996)
- T. Y. Lam, "Primeness, semiprimeness and prime radical of Ore extensions", Comm. Algebra, 8, pp.2459~2506 (1997)
- T. K. Lee, "Armendariz and reduced rings", Comm. Algebra, 32(6), pp.2287~2299 (2004)
- G. Marks, "On 2-primal Ore extensions", Comm. Algebra, 29(5), pp.2113~2123 (2001)
- G. Mason, "Reflexive ideals", Comm. Algebra, 9(17), pp.1709~1724 (1981)
- K. R. Pearson, "A skew polynomial ring over a Jacobson ring need not be a Jacobson ring", Comm. Algebra, 5(8), pp.783~794 (1977)