# Primeness is contraction-closed

*This article gives the statement (and possibly proof) of a property of ideals in commutative unital rings satisfying a metaproperty of ideals in commutative unital rings*

## Contents

## Statement

### Property-theoretic statement

The property of ideals in commutative unital rings of being a prime ideal satisfies the metaproperty of ideals in commutative unital rings of being contraction-closed.

### Verbal statement

Given a homomorphism of commutative unital rings, the contraction of a prime ideal in the ring on the right, is a prime ideal in the ring on the left.

### Symbolic statement

Suppose is a homomorphism of commutative unital rings. Then for any prime ideal of , (called the contraction of ) is a prime ideal of .

## Importance

This fact allows us to view the spectrum of a commutative unital ring as a contravariant functor, because it allows us to use a homomorphism of commutative unital rings to define a backward map , by contraction.

## Proof

*Fill this in later*