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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 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.
Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.
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 .
In other words, a projective variety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the projective coordinates ring being defined as the quotient of the graded ring or the polynomials in n + 1 variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be ...
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 ...
Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if and only if the quotient ring R/I is reduced. Prime ideals of the coordinate ring correspond to affine subvarieties.
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 ...