Palestine Journal of Mathematics Vol. 3(Spec 1) (2014) , 512–517 © Palestine Polytechnic University-PPU 2014 On Generalized Semiradical Formula Sibel Kılıçarslan Cansu and Erol Yılmaz Dedicated to Patrick Smith and John Clark on the occasion of their 70th birthdays. Communicated by Ayman Badawi MSC 2010 Classifications: 13C99, 13A99. Keywords and phrases: Semiprime Submodules, Semiradical Formula, Envelope. Abstract. Some properties of semiprime submodules are given. Semiradical of a module and the generalized semiradical formula are defined. Then we showed that Noetherian modules satisfy the generalized semiradical formula. 1 Introduction Throughout all rings are commutative and all modules are unitary. Let R be a ring and M be an R-module. A proper submodule N of M is prime if whenever rm ∈ N , for some r ∈ R, m ∈ M then m ∈ N or rM ⊆ N . A proper submodule N of an R-module M is semiprime, if whenever r k m ∈ N for some r ∈ R, m ∈ M and k ∈ Z + , then rm ∈ N . The envelope of N in M is defined as the set E M (N )= {rm : r ∈ R, m ∈ M and r k m ∈ N for some k ∈ Z + }. Semiprime submodules can be defined in terms of their envelopes, that is, a proper submod- ule N is semiprime if and only if 〈E M (N )〉 = N . This characterization can be used to show that semiprime submodules need not be prime despite the fact that every prime submodule is semiprime. For example, if R = Q[x, y, z], M = R 3 and N = 〈ze 1 ,ye 1 , xye 2 , xye 3 , xze 2 + x 2 ze 3 〉, then by [6] Theorem 2.5, 〈E M (N )〉 = N . Hence N is a semiprime submodule of M with N : M = 〈xy〉. On the other hand N is not a prime submodule; if we take r = z and m =(0, x, x 2 ), then rm = z(0, x, x 2 )=(0,xz,x 2 z) ∈ N but r = z / ∈ N : M and m =(0, x, x 2 ) / ∈ N . If N is a proper submodule of an R-module M , then the prime radical of N , rad M (N ), is the intersection of all prime submodules containing N . The semiradical of N , denoted by srad M (N ), is defined as the intersection of all semiprime submodules of M containing N . If there is no semiprime submodule containing N , then srad M (N )= M . We shall denote the semiradical of M by srad M (0). Since rad M (N ) is semiprime, we have N ⊆〈E M (N )〉⊆ srad M (N ) ⊆ rad M (N ). In section 2, we study some properties of semiprime submodules. In section 3, semiprime radical is defined and it is shown that for domains, the study of semiprime radical of any mod- ules reduces to torsion modules. Also the equality srad M (N )= 〈E M (N )〉 is investigated for some special cases. In section 4, we define generalized semiradical formula and showed that Noetherian modules satisfy the generalized semiradical formula. 2 Semiprime Submodules If N is a prime submodule of an R-module M , then it is well known that N : M is a prime ideal. If N is semiprime, we have the following. Lemma 2.1. If N is a semiprime submodule of an R-module M , then N : M is a semiprime ideal. Proof. Let x ∈ √ N : M . Then x k M ⊆ N for some k ∈ Z + . Since N is semiprime, xM ⊆ N . Hence √ N : M = N : M . This implies that N : M is a radical ideal which means that N : M is a semiprime ideal.