Extension and contraction of ideals

In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideals.

Extension of an ideal

Let A and B be two commutative rings with unity, and let f : A → B be a (unital) ring homomorphism. If $\mathfrak{𝔞}$ is an ideal in A, then $f(\mathfrak{𝔞})$ need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). The extension ${\mathfrak{𝔞}}^e$ of $\mathfrak{𝔞}$ in B is defined to be the ideal in B generated by $f(\mathfrak{𝔞})$. Explicitly,

${\mathfrak{𝔞}}^e=\{\sum y_{i}f(x_i):x_i\in \mathfrak{𝔞},y_i\in B\}$

Contraction of an ideal

If $\mathfrak{𝔟}$ is an ideal of B, then $f^{-1}(\mathfrak{𝔟})$ is always an ideal of A, called the contraction ${\mathfrak{𝔟}}^c$ of $\mathfrak{𝔟}$ to A.

Properties

Assuming f : A → B is a unital ring homomorphism, $\mathfrak{𝔞}$ is an ideal in A, $\mathfrak{𝔟}$ is an ideal in B, then:

• $\mathfrak{𝔟}$ is prime in B $\Rightarrow$ ${\mathfrak{𝔟}}^c$ is prime in A.
• ${\mathfrak{𝔞}}^{ec}\supseteq \mathfrak{𝔞}$
• ${\mathfrak{𝔟}}^{ce}\subseteq \mathfrak{𝔟}$

It is false, in general, that $\mathfrak{𝔞}$ being prime (or maximal) in A implies that ${\mathfrak{𝔞}}^e$ is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding $\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{Z}}\left[i\right]$. In $B=\mathbb{\mathbb{Z}}\left[i\right]$, the element 2 factors as $2=(1+i)(1-i)$ where (one can show) neither of $1+i,1-i$ are units in B. So $(2{)}^e$ is not prime in B (and therefore not maximal, as well). Indeed, $(1\pm i{)}^2=\pm 2i$ shows that $(1+i)=((1-i)-(1-i{)}^2)$, $(1-i)=((1+i)-(1+i{)}^2)$, and therefore $(2{)}^e=(1+i{)}^2$.

On the other hand, if f is surjective and $\mathfrak{𝔞}\supseteq \mathrm{k}\mathrm{e}\mathrm{r}f$ then:

• ${\mathfrak{𝔞}}^{ec}=\mathfrak{𝔞}$ and ${\mathfrak{𝔟}}^{ce}=\mathfrak{𝔟}$.
• $\mathfrak{𝔞}$ is a prime ideal in A $\iff$ ${\mathfrak{𝔞}}^e$ is a prime ideal in B.
• $\mathfrak{𝔞}$ is a maximal ideal in A $\iff$ ${\mathfrak{𝔞}}^e$ is a maximal ideal in B.

Extension of prime ideals in number theory

Let K be a field extension of L, and let B and A be the rings of integers of K and L, respectively. Then B is an integral extension of A, and we let f be the inclusion map from A to B. The behaviour of a prime ideal $\mathfrak{𝔞}=\mathfrak{𝔭}$ of A under extension is one of the central problems of algebraic number theory.

See also

• Splitting of prime ideals in Galois extensions

References

• Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Perseus Books, 1969,
• Math.Stackexchange, The geometric interpretation for extension of ideals?, https://math.stackexchange.com/questions/866577/the-geometric-interpretation-for-extension-of-ideals