ON LEFT O-PRIME IDEALS OVER A NONCOMMUTATIVE RING
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In this paper, we focus on a one-sided generalization of the concept of prime ideal in a noncommutative ring, which is called a left O-prime ideal. Some of its basic properties are investigated, pointing out both similarities and di¤erences between left O-prime ideals and their commutative counterparts. Mainly, we prove a noncommutative generalization of Cohen’s Theorem for left O-prime ideals and that any left ideal in R is the intersection of a nite number of left O-prime ideals of a noncommutative ring R satisfying the ascending chain condition on left O-radical ideals.
Abstract
In this paper, we focus on a one-sided generalization of the concept of prime ideal in a noncommutative ring, which is called a left O-prime ideal. Some of its basic properties are investigated, pointing out both similarities and di¤erences between left O-prime ideals and their commutative counterparts. Mainly, we prove a noncommutative generalization of Cohen’s Theorem for left O-prime ideals and that any left ideal in R is the intersection of a nite number of left O-prime ideals of a noncommutative ring R satisfying the ascending chain condition on left O-radical ideals.
참고문헌 (17)
- M. Alkan, "On prime submodules", Rocky Mountain J. Math, 37(3), pp.709~722 (2007)
- M. Alkan, "Projective modules and prime submodules", Czechoslovak Math. J, 56(131), pp.601~611 (2006)
- M.L. Reyes, "One-sided prime ideals in noncommutative algebra", Doctor, pp.2010
- W. Anderson, "Rings and Categories of Modules" (1992), Springer-Verlag
- B. J. Gardner, "Radical Theory of Rings" (2004), Marcel Dekker, Inc
- S. Çeken, "On Prime Submodules And Primary Decompositions In Two-Generated Free Modules", Taiwan. Jour. Math, 17, pp.133~142 (2013)
- S. Çeken, "On extending modules", Med.J.Math, 9, pp.129~142 (2012)
- D. S. Dummit, "Abstract Algebra" (1999), Prentice Hall
- J. Dauns, "Prime modules", J. Reine Angew Math, 298, pp.156~181 (1978)
- T. Y. Lam, "A First Course in Noncommutative Rings" (2001), Springer
- J.C. McConnel, "Noncommutative Noetherian Rings" (1987), Wiley Chichester
- P. F. Smith, "Radical submodules and uniform dimension of modules", Turk J. Math, 28, pp.255~270 (2004)
- J. Dauns, "Prime modules and onesided ideals in ring Theory and Algebra III", Proceeding of the Third Oklahoma Conference, pp.301~344 (1980)
- K.H. Leung, "On Commutative Noetherian Rings which satisfy the radical formula", Glasgow Math J, 39, pp.285~293 (1997)
- Y. Tiras, "Prime modules and submodules", Comm. in Algebra, 31, pp.395~396 (2003)
- Irving Kaplansky, "Commutative Rings" (1974), University of Chicago Press
- I. S. Cohen, "Commutative rings with restricted minimum condition", Duke Math. J, 17, pp.27~42 (1950)