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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.
The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b , then c n = a n + b n for every n in N .
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.
Spec k[t], the spectrum of the polynomial ring over a field k: such a polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k[t]. If k is algebraically closed , for example the field of complex numbers , a non-constant polynomial is irreducible if and only if it is linear, of the form ...
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 .
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 ...
If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring [, …,] / is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring. Projective varieties arise in many ways.