Extension and Contraction

Let \(f: R\rightarrow S\) be a ring homomorphism. Let \(I \triangleleft R, J \triangleleft S\). The extension \(I^e\) of \(I\) (with respect to \(f\)) is \(\langle f(I) \rangle\). Then contraction \(J^c\) of \(J\) (with respect to \(f\)) is \(f^{-1}(J)\), which is an ideal in \(R\).

Note that if \(J\) is prime than so is \(J^c\), though the same is not always true for extensions. For example, take the identity map \(f:\mathbb{Z} \rightarrow \mathbb{Q}\). Then for any prime \(p\), \(p\mathbb{Z}\) is prime in \(\mathbb{Z}\) but \((p\mathbb{Z})^e = \mathbb{Q}\) which is not prime in \(\mathbb{Q}\).

In general there is no simple relationship between the prime ideals of \(R\) and \(S\). For example consider the identity map \(f:\mathbb{Z}\rightarrow\mathbb{Z}[i]\) (see notes on <a href="../numberfield"> number fields</a>).

Other properties of extension and contraction:

  1. \(I\subset I^{e c}, J\supset J^{c e}\)

  2. \(J^{c} = J^{c e c}, I = I^{e c e}\)

  3. Let \(\mathcal{C}\) be the set of contracted ideals in \(R\) and \(\mathcal{E}\) be the set of extended ideals in \(S\). Then \(\mathcal{C} = \{ K\triangleleft R | K^{e c} = K\}\), \(\mathcal{E} = \{ L\triangleleft S| L^{c e} = L\}\), and \(K \mapsto K^{e}\) for all \(K \in \mathcal{C}\) defines a bijection \(\mathcal{C} \rightarrow \mathcal{E}\) whose inverse is \(L \mapsto L^{c}\) for all \(L \in \mathcal{E}\).

Let \(I_1,I_2\triangleleft R\) and \(J_1,J_2 \triangleleft S\). Then

  1. \((I_1 + I_2)^e = I_1^e + I_2^e\), \((J_1 + J_2)^c \supset J_1^c + J_2^c\)

  2. \(I_1 \cap I_2)^e \subset I_1^e \cap I_2^e\), \((J_1 \cap J_2)^c = J_1^c \cap J_2^c\)

  3. \((I_1 I_2)^e = I_1^e I_2^e\), \((J_1 J_2)^c \supset J_1^c J_2^c\)

  4. \((I_1 :I_2)^e \subset (I_1^e :I_2^e)\), \((J_1 :J_2)^c \subset (J_1^c :J_2^c)\)

  5. \((\sqrt{I_1})^e \subset \sqrt{I_1^e}\), \((\sqrt{J_1})^c = \sqrt{J_1^c}\)

The set \(\mathcal{E}\) is closed under sum and product while the set \(\mathcal{C}\) is closed under intersection, forming ideal quotients and taking radicals.

Ben Lynn blynn@cs.stanford.edu 💡