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

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 $f:R\to S$ is a homomorphism of commutative unital rings. Then for any prime ideal $I$ of $S$, $I^{c}=f^{-1}(I)$ (called the contraction of $I$) is a prime ideal of $R$.

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 $f:R\to S$ to define a backward map $Spec(f):Spec(S)\to Spec(R)$, by contraction.

Proof

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