Beck's monadicity theorem

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, ISBN 0-201-00361-9