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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.
If R denotes the ring [,] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve). In the ring Z [ X ] {\displaystyle \mathbb {Z} [X]} of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal.
Given a polynomial p of degree d, the quotient ring of K[X] by the ideal generated by p can be identified with the vector space of the polynomials of degrees less than d, with the "multiplication modulo p" as a multiplication, the multiplication modulo p consisting of the remainder under the division by p of the (usual) product of polynomials.
A polynomial ring in infinitely many variables: ... is a prime ideal of a commutative ring ... The ring structure in cohomology provides the foundation for ...
Now, for any commutative ring R, an ideal I and a minimal prime P over I, the pre-image of I R P under the localization map is the smallest P-primary ideal containing I. [18] Thus, in the setting of preceding theorem, the primary ideal Q corresponding to a minimal prime P is also the smallest P -primary ideal containing I and is called the P ...
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 ...
The dimension of the graded ring = / + (equivalently, 1 plus the degree of its Hilbert polynomial). A commutative ring R is said to be catenary if for every pair of prime ideals ′, there exists a finite chain of prime ideals = = ′ that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals ...
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 .