On Jacobson and nil radicals related to polynomial rings
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that $J(R[x])=J(R)[x]$ if and only if $J(R)$ is nil when a given ring $R$ is Armendariz, where $J(A)$ means the Jacobson radical of a ring $A$. A ring will be called {\it feckly Armendariz} if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.
Abstract
This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that $J(R[x])=J(R)[x]$ if and only if $J(R)$ is nil when a given ring $R$ is Armendariz, where $J(A)$ means the Jacobson radical of a ring $A$. A ring will be called {\it feckly Armendariz} if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.
참고문헌 (31)
- S. A. Amitsur, "A general theory of radicals III", Amer. J. Math, 76, pp.126~136 (1954)
- S. A. Amitsur, "Radicals of polynomial rings", Canad. J. Math, 8, pp.355~361 (1956)
- D. D. Anderson, "Armendariz rings and Gaussian rings", Comm. Algebra, 26(7), pp.2265~2272 (1998)
- R. Antoine, "Nilpotent elements and Armendariz rings", J. Algebra, 319(8), pp.3128~3140 (2008)
- R. Antoine, "Examples of Armendariz rings", Comm. Algebra, 38(11), pp.4130~4143 (2010)
- E. P. Armendariz, "A note on extensions of Baer and P.P.-rings", J. Aust. Math. Soc, 18, pp.470~473 (1974)
- V. Camillo, "Nilpotent ideals in polynomial and power series rings", Proc. Amer. Math. Soc, 138(5), pp.1607~1619 (2010)
- J. L. Dorroh, "Concerning adjunctions to algebras", Bull. Amer. Math. Soc, 38(2), pp.85~88 (1932)
- K. E. Eldridge, "Orders for finite noncommutative rings with unity", Amer. Math. Monthly, 73, pp.512~514 (1966)
- K. R. Goodearl, "Von Neumann Regular Rings" (1979), Pitman
- K. R. Goodearl, "An Introduction to Noncommutative Noetherian Rings" (1989), Cambridge University Press
- 한준철, "Armendariz property over prime radicals", 대한수학회지, 50(5), pp.973~989 (2013), 대한수학회
- J. C. Han, "Extensions of clean rings", Comm. Algebra, 29(6), pp.2589~2595 (2001)
- Y. Hirano, "On rings whose prime radical contains all nilpotent elements of index two", Arch. Math, 66(5), pp.360~365 (1996)
- S. U. Hwang, "Structure and topological conditions of NI rings", J. Algebra, 302(1), pp.186~199 (2006)
- Young Cheol Jeon, "On weak Armendariz rings", 대한수학회보, 46(1), pp.135~146 (2009), 대한수학회
- D. W. Jung, "Nil-Armendariz rings and upper nilradicals", Internat. J. Algebra Comput, 22(6), pp.13~ (2012)
- N. K. Kim, "Power series rings satisfying a zero divisor property", Comm. Algebra, 34(6), pp.2205~2218 (2006)
- N. K. Kim, "On right quasi-duo rings which are π-regular", Bull. Korean Math. Soc, 37(2), pp.217~227 (2000)
- N. K. Kim, "Armendariz rings and reduced rings", J. Algebra, 223(2), pp.477~488 (2000)
- J. Lambek, "Lectures on Rings and Modules" (1966), Blaisdell Publishing Company
- C. Lanski, "Nil subrings of Goldie rings are nilpotent", Canad. J. Math, 21, pp.904~907 (1969)
- T.-K. Lee, "On Armendariz rings", Houston J. Math, 29(3), pp.583~593 (2003)
- C. Levitzki, "A theorem on polynomial identities", Proc. Amer. Math. Soc, 1, pp.331~333 (1950)
- A. R. Nasr-Isfahani, "A generalization of reduced rings", J. Algebra Appl, 11(4), pp.30~ (2012)
- W. K. Nicholson, "Lifting idempotents and exchange rings", Trans. Amer. Math. Soc, 229, pp.269~278 (1977)
- M. B. Rege, "Armendariz rings", Proc. Japan Acad. Ser. A Math. Sci, 73(1), pp.14~17 (1997)
- A. Smoktunowicz, "Polynomial rings over nil rings need not be nil", J. Algebra, 233(2), pp.427~436 (2000)
- J. Stock, "On rings whose projective modules have the exchange property", J. Algebra, 103(2), pp.437~453 (1986)
- B. Ungor, "Feckly reduced rings", Hacettepe J. Math. Stat, 44, pp.375~384 (2015)
- R. B. Warfield, "Exchange rings and decompositions of modules", Math. Ann, 199, pp.31~36 (1972)