Primeness is contraction-closed

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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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