Abstract
Let R be any ring; a ∈ R is called a weak zero-divisor if there are r, s ∈ R with ras = 0 and rs = 0. It is shown that, in any ring R, the elements of a minimal prime ideal are weak zero-divisors. Examples show that a minimal prime ideal may have elements which are neither left nor right zero-divisors. However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors. The union of the minimal prime ideals is studied in 2-primal rings and the union of the minimal strongly prime ideals (in the sense of Rowen) in NI-rings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G.K. Birkenmeier, J.Y. Kim and J.K. Park, Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), 213–230.
G.K. Birkenmeier, J.Y. Kim and J.K. Park, A characterization of minimal prime ideals, Glasgow Math. J. 40 (1998), 223–236.
W.D. Burgess and W. Stephenson, Pierce sheaves of non-commutative rings, Comm. Algebra 4 (1976), 51–75.
K.R. Goodearl, Von Neumann Regular Rings. Krieger, 1991.
C.Y. Hong and T.K. Kwak, On minimal strongly prime ideals, Comm. Algebra 28 (2000), 4867–4878.
S.U. Hwang, Y.C. Jeon and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), 186–199.
K.-H. Kang, B.-O. Kim, S.-J. Nam and S.-H. Sohn, Rings whose prime radicals are completely prime, Commun. Korean Math. Soc. 20 (2005), 457–466.
P.T. Johnstone, Stone Spaces. Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982.
A. Kaučikas and R. Wisbauer, On strongly prime rings and ideals, Comm. Algebra 28 (2000), 5461–5473.
T.Y. Lam, Lectures on Modules and Rings. Springer-Verlag, 1998.
T.Y. Lam, A First Course in Ring Theory. Second Edition, Springer-Verlag, 2001.
J. Lambek, Lectures on Rings and Modules. Chelsea Publication Co., 1986.
L.H. Rowen, Ring Theory. Student Edition, Academic Press, 1991.
G. Shin, Prime ideals and sheaf representations of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43–60.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to S.K. Jain on his seventieth birthday
Rights and permissions
Copyright information
© 2010 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Burgess, W.D., Lashgari, A., Mojiri, A. (2010). Elements of Minimal Prime Ideals in General Rings. In: Van Huynh, D., López-Permouth, S.R. (eds) Advances in Ring Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0286-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0286-0_6
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0285-3
Online ISBN: 978-3-0346-0286-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)