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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.
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.
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 .
The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings. [14] Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in , if a n is in I for some n, then a is in I.
The Zariski topology defines a topology on the spectrum of a ring (the set of prime ideals). [2] In this formulation, the Zariski-closed sets are taken to be the sets = {()} where A is a fixed commutative ring and I is an ideal. This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined ...
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 ...
Since V is assumed to be a variety, and so an irreducible algebraic set, the ideal I can be chosen to be a prime ideal, and so R is an integral domain.The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero.
In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed. Let = [, …,] be a polynomial ring over a field k. An ideal I in R is called monomial if it is generated by monomials; i.e., . The integral closure of a monomial ideal is monomial.