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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.
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 .
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.
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 ...
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.
The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to R / x R {\displaystyle R/xR} for some x ∈ R {\displaystyle x\in R} [ 4 ] (notice that x {\displaystyle x} may be equal ...
Every prime ideal P of R such that R/P has an element x with (R/P)[x −1] a field is a maximal prime ideal. The spectrum of R is a Jacobson space, meaning that every closed subset is the closure of the set of closed points in it. (For Noetherian rings R): R has no prime ideals P such that R/P is a 1-dimensional semi-local ring.