An associated prime of an $R$-module $M$ is an ideal of the form $Ann_R(N)$ where $N$ is a prime sub-module of $M$ in the sense that $N$ is nonzero and $Ann_R(N)=Ann_R(N')$ for each nonzero sub-module $N'$ of $N$. It is known that an associated prime is a prime ideal.
My question is that if $M$ is an ideal of $R$ then is the above-mentioned equivalent to the associated prime ideals arisen from primary decomposition?
Thanks!
It seems you use the definition of associated primes given by Lam in Lectures on Modules and Rings, page 86. If your question refers to commutative rings, then on the same page Lam proves that for commutative rings the above definition of associated primes coincides with the usual definition (as annihilators of an element), and then the theory says that your guess is true.
(I'd rise the same question for $M$ an $R$-module. Modules can also have primary decompositions. Why should restrict us to ideals?)