# 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 $$(I:\sum_{l\in X} J_l) = \cap_{l\in X}(I:J_l)$$

Ben Lynn blynn@cs.stanford.edu 💡