Sometimes one considers the maximal spectrum , which is the subspace of consisting of the closed points. In this paper, we investigate connections between some algebraic properties of commutative rings and topological properties of their minimal and maximal prime spectrum with respect to the flat topology. As a set, we define the spectrum of R, written Spec(R), to ... at a point x corresponding to a maximal ideals M lives in the quotient ring A(X)/M, and all of these quotients can be canonically identified with the field k. Lemma Let R be a commutative ring, f ∈ R an element, and P ⊂ R a prime ideal. Also, we give a characterization for rings whose maximal prime spectrum is a compact topological space with respect to the flat topology.
For a graded ring one also considers the projective spectrum . If A is the zero ring, there is no proper prime ideal of A: Hence SpecA is empty.
Active 7 years ago. In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals.The Krull dimension need not be finite even for a Noetherian ring.More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. In these notes we will be working always in the category of commutative rings with unity. Which commutative rings have irreducible (maximal) spectra? A ring R is said to have Noetherian spectrum if R satisfies the ascending chain condition (ACC) on radical ideals, and R is said to have Noetherian maximal spectrum if R satisfies the ACC on J-radical ideals. Math.
Similarly, Z₆/<3> ≅ Z₃ is is an integral domain and a field (being of prime order).
As we will discuss in Sect.4, for many applications it is preferred to have a relatively large FSR (several nm), and this implies the use of small rings. In this article, we investigate the interplay between the topological properties of \(Max_{R}(M)\) and module theoretic properties of M.Also, for various types of modules M, we obtain some conditions under which \(Max_{R}(M)\) is homeomorphic …
If A is the zero ring, there is no proper prime ideal of A: Hence SpecA is empty. $\begingroup$ Armando j18eos, you are right, there is many rings with noetherian mix-spectrum, but I am looking for a condition $*$ that ($*$ +noetherian max-spectrum) implies (noetherian spectrum). Sis a local ring with maximal ideal (x;y), the only maximal ideal in the ring [This idea was told to me by Umang Varma.]
On the maximal spectrum of commutative semiprimitive rings K. Samei. However, there has not been an example of such a commutative ring yet. Then f(P)=0 if and only if f ∈ P. Proof: Obvious. This allows you to study rings by studying the geometry of certain spaces.
Thus, <3> is both prime and maximal. Sometimes one considers the maximal spectrum , which is the subspace of consisting of the closed points. The nilradical is prime precisely when the spectrum is irreducible. If A is a nonzero ring, A has a maximal ideal (by Zorn’s lemma) (or every ideal is contained in a maximal ideal we can choose the zero ideal). In fact, every prime ideal is maximal in a commutative ring whenever there exists an …
The preimage of a prime ideal under a ring homomorphism is a prime ideal.
It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if RP is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. Ask Question Asked 7 years ago. The set of all prime ideals (the spectrum of a ring) contains minimal elements (called minimal prime). Comments: The projective modulus of a (commutative) ring is defined and a class of quotient rings is given for which the projective moduli are arbitrarily smaller than the dimension of the maximal spectra.
Proposition 1.1.
Thus, <2> is both prime and maximal.
Geometrically, these correspond to irreducible components of the spectrum.
If , then the points of are the homogeneous prime ideals of such that . In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. In this paper, Zariski compactness of the minimal spectrum and flat compactness of the maximal spectrum are characterized. \(Max_{R}(M)\)) of M is the collection of all prime (resp.
Then SpecA is nonempty. It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum.
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