Ideal Quotients

Let $R$ be a ring. Let $I, J \triangleleft R$. The ideal quotient of $I$ by $J$ is

\[ (I:J) =\{x\in R|J x \subset I\} \]

We call

\[ (0:J)=(\{0\}:J) =\{x\in R|J x =\{0\}\} \]

the annihilator of $J$, also denoted by $Ann(J)$.

If $y\in R$ we write $(I:y) = (I:R y)$ and $Ann(y) = Ann(A y)$. Using this notation, we see that the set of zero divisors of $R$ is precisely the set $\cup_{x\ne 0} Ann(x)$.

Example: Take $R=\mathbb{Z}$. Let $m,n \in \mathbb{Z}^+$. Write $m = p_1^{\alpha_1}...p_k^{\alpha_k}, n = p_1^{\beta_1}...p_k^{\beta_k}$ for some primes $p_1,...,p_k$ and nonnegative exponents $\alpha_1,...\alpha_k, \beta_1,...,\beta_k$. Then

\[ (m\mathbb{Z} : n) = \{ z\in \mathbb{Z} | zn \in m\mathbb{Z}\} =q\mathbb{Z} \]

where $q = p_1^{\gamma_1}...p_k^{\gamma_k}$ and

\[ \gamma_i = max\{\alpha_i -\beta_i, 0\} = \alpha_i - min\{\alpha_i, \beta_i\} \]

In other words, $(m\mathbb{Z}:n) = q\mathbb{Z}$ where $q = m /gcd(m,n)$.

The following are easily verified.

  1. $I \subset (I:J)$

  2. $(I:J)J \subset I$

  3. $((I:J):K)=(I:J K)=((I:K):J)$

  4. If $I_l\triangleleft R$ for $l\in X$ then $(\cap_{l\in X} I_l : J) = \cap_{l\in X}(I_l :J)$

  5. If $J_l\triangleleft R$ for $l\in X$ then $(I:\sum_{l\in X} J_l) = \cap_{l\in X}(I:J_l)$