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 💡