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(The zero ring has no prime ideals, because the ideal (0) is the whole ring.) An ideal I is prime if and only if its set-theoretic complement is multiplicatively closed. [3] Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of Krull's theorem.
Prime ideal: A proper ideal is called a prime ideal if for any and in , if is in , then at least one of and is in . The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig.
A proper ideal P of R is called a prime ideal if for any elements , we have that implies either or . Equivalently, P is prime if for any ideals I, J we have that IJ ⊆ P implies either I ⊆ P or J ⊆ P. This latter formulation illustrates the idea of ideals as generalizations of elements.
An affine algebraic variety corresponds to a prime ideal in a polynomial ring, and the points of such an affine variety correspond to the maximal ideals that contain this prime ideal. The Zariski topology , originally defined on an algebraic variety, has been extended to the sets of the prime ideals of any commutative ring; for this topology ...
Every ring homomorphism: induces a continuous map (): (since the preimage of any prime ideal in is a prime ideal in ). In this way, Spec {\displaystyle \operatorname {Spec} } can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces .
If X is an affine algebraic set, and I is the ideal of all polynomials that are zero on X, then the quotient ring = [, …,] / is called the coordinate ring of X. If X is an affine variety, then I is prime, so the coordinate ring is an integral domain .
4. prime ring : A nonzero ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we have either a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every simple ring and every domain is a prime ring. primitive 1.