Armendariz rings and Gaussian rings. (English) Zbl 0915.13001

Anderson, D. D.; Camillo, Victor

Commun. Algebra 26, No. 7, 2265-2272 (1998).

The authors establish the following results:
(1) A ring \(R\) is Armendariz \(\Leftrightarrow R[x]\) is Armendariz.
(2) Let \(R\) be a von Neumann ring. Then \(R\) is Armendariz \(\Leftrightarrow R\) is reduced.
(3) Let \(R\) be a commutative ring. Then \(R\) is Gaussian \(\Leftrightarrow\) every homomorphic image of \(R\) is Armendariz.
(4) Let \(R\) be a commutative rings. Then \(R[x]\) is Gaussian \(\Leftrightarrow R\) is von Neumann regular.
(5) For a commutative ring \(R\), the power series ring \(R[[x]]\) is Gaussian \(\Leftrightarrow R[[x]]\) is a reduced arithmetical ring \(\Leftrightarrow R[[x]]\) has weak global dimension one \(\Leftrightarrow R[[x]]\) is Bezout \(\Leftrightarrow R\) is von Neumann regular and \(x_0\)-algebraically compact.

Reviewer: K.Chandrasekhara Rao (Karaikudi)

MSC:

13A15	Ideals and multiplicative ideal theory in commutative rings
16E50	von Neumann regular rings and generalizations (associative algebraic aspects)
13F25	Formal power series rings
13F20	Polynomial rings and ideals; rings of integer-valued polynomials
16S36	Ordinary and skew polynomial rings and semigroup rings

Keywords:

Gaussian ring; Armendariz rings; power series ring; von Neumann regular ring

References:

[1]	DOI: 10.1016/0021-8693(77)90318-0 · Zbl 0406.13004 · doi:10.1016/0021-8693(77)90318-0
[2]	DOI: 10.1006/jabr.1996.0110 · Zbl 0857.13017 · doi:10.1006/jabr.1996.0110
[3]	DOI: 10.1017/S1446788700029190 · Zbl 0292.16009 · doi:10.1017/S1446788700029190
[4]	DOI: 10.1016/0021-8693(77)90405-7 · Zbl 0393.13004 · doi:10.1016/0021-8693(77)90405-7
[5]	DOI: 10.1090/S0002-9939-1986-0861751-0 · doi:10.1090/S0002-9939-1986-0861751-0
[6]	Gilmer R., Queen’s Papers in Pure and Applied Mathematics 90 (1992)
[7]	Gilmer R., J. Reine Angew. Math 278 pp 145– (1975)
[8]	Goodearl K.R., Von Neumann Regular Rings (1991)
[9]	DOI: 10.1090/S0002-9939-1964-0179197-5 · doi:10.1090/S0002-9939-1964-0179197-5
[10]	DOI: 10.3792/pjaa.73.14 · Zbl 0960.16038 · doi:10.3792/pjaa.73.14
[11]	Tsang H., Dissertation (1965)