Let $R$ be a commutative ring with $1$. We know that the *Krull dimension of $R$* is by definition the length of the longest chain of prime ideals of $R$.

Now if $M$ is a $R$-module, the *Krull dimension of $M$* is by definition $\dim(M):=\dim(R/\mathrm{Ann}_R(M))$. Since every ideal $I$ of $R$ is also a $R$-module, the Krull dimension of $I$ is $\dim(I)=\dim(R/\mathrm{Ann}_R(I))$.

However, in the literature, the Krull dimension of an ideal is $\dim(I):=\dim(R/I)$.

**Are the two definitions equivalent?**