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 $𝔞$ is an ideal in A, then $f (𝔞)$ 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 $𝔞^{e}$ of $𝔞$ in B is defined to be the ideal in B generated by $f (𝔞)$ . Explicitly,

𝔞^{e} = {\sum y_{i} f (x_{i}) : x_{i} \in 𝔞, y_{i} \in B}

Contraction of an ideal

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

Properties

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

$𝔟$ is prime in B $\Rightarrow$ $𝔟^{c}$ is prime in A.

$𝔞^{e c} \supseteq 𝔞$

$𝔟^{c e} \subseteq 𝔟$

- It is false, in general, that $𝔞$ being prime (or maximal) in A implies that $𝔞^{e}$ is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding $ℤ \to ℤ [i]$ . In $B = ℤ [i]$ , 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, $(1 \pm i)^{2} = \pm 2 i$ 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 $𝔞 \supseteq k e r f$ then:

$𝔞^{e c} = 𝔞$ and $𝔟^{c e} = 𝔟$ .

$𝔞$ is a prime ideal in A $\Leftrightarrow$ $𝔞^{e}$ is a prime ideal in B.

$𝔞$ is a maximal ideal in A $\Leftrightarrow$ $𝔞^{e}$ 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 $𝔞 = 𝔭$ of A under extension is one of the central problems of algebraic number theory.

References

Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9

Beck's monadicity theorem

Contents

Extension of an ideal

Contraction of an ideal

Properties

Extension of prime ideals in number theory

See also

References

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Beck's monadicity theorem

Extension of an ideal

Contraction of an ideal

Properties

Extension of prime ideals in number theory

See also

References

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