One dimensional local domains and radical formula

  • Anand ParkashEmail author
Original Paper


If \((R, P)\) is a local domain of dimension one, then it is shown that \(R\) satisfies the radical formula if and only if \(( Ra _1: Ra _2+ Ra _3+\cdots + Ra _n)=P( Ra _1: Ra _2+ Ra _3+\cdots + Ra _n)\) for every integer \(n\ge 2\) and for every \(a_1, a_2,\ldots ,a_n\in R\) such that \(a_i\notin Ra _j\) for every \(i\ne j\).


Prime submodules Radical formula Local rings Integral domains 

Mathematics Subject Classification (1991)

13C13 13C99 


  1. Jenkins, J., Smith, P.F.: On the prime radical of a module over a commutative ring. Commun. Algebra 20, 3593–3602 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Leung, K.H., Man, S.H.: On commutative Noetherian rings which satisfy the radical formula. Glasgow Math. J. 39, 285–293 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Man, S.H.: One dimensional domains which satisfy the radical formula are Dedekind domains. Arch. Math. 66, 276–279 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Man, S.H.: On commutative Noetherian rings which satisfy the generalized radical formula. Commun. Algebra 27, 4075–4088 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. McCasland, R.L., Moore, M.E.: On radicals of submodules. Commun. Algebra 19, 1327–1341 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Parkash, A.: Arithmetical rings satisfy the radical formula. J. Commut. Algebra 4, 293–296 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2014

Authors and Affiliations

  1. 1.Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia

Personalised recommendations