A right R -module M will be called a second module provided M ?0 and ann ( M ) = ann ( M/N ) for every proper submodule N of M . By a second submodule of a module,we mean a submodule which is also a second module. A non-zero R -module M is called a prime module if ann ( M ) = ann ( K ) for every non-zero sub-module K of M . A proper submodule N of a module M is called a prime submodule of M if M/N is a prime module.The intersection of all prime submodules of a module M is called the prime radical of M and is denoted by rad( M ). If there is no prime submodule in M, then rad( M ) = M . The second radical, Sec( M ), of a module M is defined to be the sum of all the second submodules of M .