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A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R - module , and P is a maximal element in the poset of annihilator ideals of submodules ...
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig.
Prime ideal: A proper ideal is called a prime ideal if for any and in , if is in , then at least one of and is in . The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
By definition, an associated prime is a prime ideal which is the annihilator of a nonzero element of M; that is, = for some (this implies ). Equivalently, a prime ideal p {\displaystyle {\mathfrak {p}}} is an associated prime of M if there is an injection of R -modules R / p ↪ M {\displaystyle R/{\mathfrak {p}}\hookrightarrow M} .
A polynomial ring in infinitely many variables: ... is a prime ideal of a commutative ring ... The ring structure in cohomology provides the foundation for ...
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 ...
By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely.
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 .