PRIME RADICALS OF SKEW LAURENT POLYNOMIAL RINGS

Title & Authors
PRIME RADICALS OF SKEW LAURENT POLYNOMIAL RINGS
Han, Jun-Cheol;

Abstract
Let R be a ring with an automorphism 17. An ideal [ of R is ($\small{\sigma}$-ideal of R if $\small{\sigma}$(I).= I. A proper ideal P of R is ($\small{\sigma}$-prime ideal of R if P is a $\small{\sigma}$-ideal of R and for $\small{\sigma}$-ideals I and J of R, IJ $\small{\subseteq}$ P implies that I $\small{\subseteq}$ P or J $\small{\subseteq}$ P. A proper ideal Q of R is $\small{\sigma}$-semiprime ideal of Q if Q is a $\small{\sigma}$-ideal and for a $\small{\sigma}$-ideal I of R, I$\small{^{2}}$ $\small{\subseteq}$ Q implies that I $\small{\subseteq}$ Q. The $\small{\sigma}$-prime radical is defined by the intersection of all $\small{\sigma}$-prime ideals of R and is denoted by P$\small{_{}$(R). In this paper, the following results are obtained: (1) For a principal ideal domain R, P$\small{_{}$(R) is the smallest $\small{\sigma}$-semiprime ideal of R; (2) For any ring R with an automorphism $\small{\sigma}$ and for a skew Laurent polynomial ring R[x, x$\small{^{-1}}$; $\small{\sigma}$], the prime radical of R[x, x$\small{^{-1}}$; $\small{\sigma}$] is equal to P$\small{_{}$(R)[x, x$\small{^{-1}}$; $\small{\sigma}$ ].
Keywords
sigma-semiprime ring;sigma-prime ring;sigma-prime radical;skew Laurent polynomial rin;
Language
English
Cited by
1.
On primeness of general skew inverse Laurent series ring, Communications in Algebra, 2017, 45, 3, 919
2.
Some Results On Prime Skew Rings, Communications in Algebra, 2012, 40, 2, 779
References
1.
A. W. Goldie and G. O. Michler, Ore extensions and polycyclic group rings, J. London. Math. Soc. (2) 9 (1974), 337-345

2.
D. A. Jordan, Primitive skew Laurent polynomial rings, Glasg. Math. J. 19 (1978), 79-85

3.
D. A. Jordan, Primitivity in skew Laurent polynomial rings and related rings, Math. Z. 213 (1993), 353-371

4.
T. Y. Lam, A first course in noncommutative rings, Springer-Verlag, New York, 1991

5.
A. Leloy and J. Matczuk, Primitivity of skew polynomial and skew Laurent polynomial rings, Comm. Algebra 24 (1996), no. 7, 2271-2284

6.
A. Moussavi, On the semiprimitivity of skew polynomial rings, Proc. Edinb. Math. Soc. 36 (1993), 169-178