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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.
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.
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: [,, ] = ... If is a prime ideal of a commutative ring R, then the field of ...
A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal. In a Noetherian ring, every prime ideal
An ideal I in the ring R (with unity) is prime if the factor ring R/I is an integral domain. Equivalently, I is prime if whenever then either or . In an integral domain, a nonzero principal ideal is prime if and only if it is generated by a prime element.
The question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x 2 + y 3 + z 5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) the local ring is a UFD but ...
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 ...