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On modules which satisfy the radical formula

In this paper, the authors prove that every representable module over a commutative ring with identity satisfies the radical formula. With this result, they extend the class of modules satisfying the radical formula from that of Artinian modules to a larger one. They conclude their work by giving a description of the radical of a submodule of a representable module. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 195 – 201 ¨ ITAK ˙ c TUB doi:10.3906/mat-1103-9 On modules which satisfy the radical formula B¨ ulent SARAC ¸ ∗, Y¨ ucel TIRAS ¸ Department of Mathematics, Hacettepe University, 06532, Beytepe, Ankara, Turkey Received: 04.03.2011 • Accepted: 28.01.2012 • Published Online: 19.03.2013 • Printed: 22.04.2013 Abstract: In this paper, the authors prove that every representable module over a commutative ring with identity satisfies the radical formula. With this result, they extend the class of modules satisfying the radical formula from that of Artinian modules to a larger one. They conclude their work by giving a description of the radical of a submodule of a representable module. Key words: Prime submodule, prime radical, radical formula, secondary module, secondary representation, representable module 1. Introduction Throughout this work R will denote a commutative ring with identity and every module will be unitary. Let M be an R -module. For submodules K and L of M, we use the notation (K : L) to show the ideal {r ∈ R : rL ≤ K} of R. A proper submodule N of M is said to be prime submodule of M, if, for every r ∈ R and m ∈ M, rm ∈ N implies m ∈ N or r ∈ (N : M ). It is not difficult to see that if N is a prime submodule of M and P = (N : M ) then P is a prime ideal of R and, in this case, we say that N is P -prime. It is easy to see that if M = R prime ideals and prime submodules of R coincide. For any submodule N of M , the (prime) radical of N in M, denoted by radM (N ), is defined to be the intersection of all prime submodules of M containing N. (If there is no such prime submodule in M we put radM (N ) = M ). It is not easy to calculate the radical of a submodule, in general. Several authors tried to give simple descriptions for the radical in some particular cases. In this note, we shall need the notion of the envelope of a submodule introduced by