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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.
However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized ...
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 .
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 ...
Note that in an integral domain, the ideal (0) is a prime ideal, but 0 is an exception in the definition of 'prime element'.) Interest in prime elements comes from the fundamental theorem of arithmetic, which asserts that each nonzero integer can be written in essentially only one way as 1 or −1 multiplied by a product of positive prime numbers.
Gröbner bases are primarily defined for ideals in a polynomial ring = [, …,] over a field K.Although the theory works for any field, most Gröbner basis computations are done either when K is the field of rationals or the integers modulo a prime number.