Ideal Quotients
Let \(R\) be a ring. Let \(I, J \triangleleft R\). The ideal quotient of \(I\) by \(J\) is
We call
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
where \(q = p_1^{\gamma_1}...p_k^{\gamma_k}\) and
In other words, \((m\mathbb{Z}:n) = q\mathbb{Z}\) where \(q = m /gcd(m,n)\).
The following are easily verified.
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\(I \subset (I:J)\)
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\((I:J)J \subset I\)
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\(((I:J):K)=(I:J K)=((I:K):J)\)
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If \(I_l\triangleleft R\) for \(l\in X\) then \((\cap_{l\in X} I_l : J) = \cap_{l\in X}(I_l :J)\)
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If \(J_l\triangleleft R\) for \(l\in X\) then \((I:\sum_{l\in X} J_l) = \cap_{l\in X}(I:J_l)\)