Primes and primaries in module

M an R-module.
An element r∈R is a zero divisor in M if rm = 0 for some m≠0. r is a zero in M if rm = 0 ∀m∈M. r is a nilpotent in M if some positive power of r is a zero in M. Provided M≠0, zeros ⊆ nilpotents ⊆ zero divisors.
Let N⊆M be a submodule. N is a prime submodule if N≠M and every zero divisor in M/N is a zero in M/N. N is a primary submodule if N≠M and every nilpotent in M/N is a zero divisor in M/N. Then prime submodules are primary.
Let P⊆M be a prime submodule. Then (P:M) is a prime ideal.
Let Q⊆M be a primary submodule. Then (Q:M) is a primary ideal.
If one defines rad(N)=rad((N:M)) for submodules N⊆M, then rad(Q)=rad((Q:M)).