Armendariz rings and reduced rings. (English) Zbl 0957.16018
According to D. D. Anderson and V. Camillo [Commun. Algebra 26, No. 7, 2265-2272 (1998; Zbl 0915.13001)], a ring \(R\) is called Armendariz if whenever polynomials \(f(x)=a_0+a_1x+\cdots+a_m x^m\) and \(g(x)=b_0+b_1x+\cdots+b_nx^n\) in \(R[x]\) satisfy \(f(x)g(x)=0\), then \(a_ib_j=0\) for each \(i\), \(j\). It is shown that if a ring \(R\) is Armendariz, then \(R\) is PP (resp. Baer) if and only if \(R[x]\) is PP (resp. Baer). Also it is shown that a semisimple Artinian ring \(R\) is Armendariz if and only if \(R\) is a direct sum of division rings.
Reviewer: J.K.Park (Pusan)
MSC:
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |
| 16P20 | Artinian rings and modules (associative rings and algebras) |
Keywords:
Armendariz rings; reduced rings; Baer rings; PP rings; semisimple Artinian rings; direct sums of division ringsCitations:
Zbl 0915.13001References:
| [1] | Anderson, D. D.; Camillo, V., Armendariz rings and Gaussian rings, Comm. Algebra, 26, 2265-2272 (1998) · Zbl 0915.13001 |
| [2] | Armendariz, E. P., A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc., 18, 470-473 (1974) · Zbl 0292.16009 |
| [3] | Kaplansky, I., Rings of Operators (1968), Benjamin: Benjamin New York · Zbl 0212.39101 |
| [4] | Lee, Y.; Huh, C., Counterexamples on p.p.-rings, Kyungpook Math. J., 38, 421-427 (1998) · Zbl 0944.16026 |
| [5] | Lee, Y.; Kim, N. K.; Hong, C. Y., Counterexamples on Baer rings, Comm. Algebra, 25, 497-507 (1997) · Zbl 0874.16009 |
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| [7] | Rege, M. B.; Chhawchharia, S., Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci., 73, 14-17 (1997) · Zbl 0960.16038 |
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