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Any maximal ideal is a prime ideal or, more briefly, is prime. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. This ring has only one maximal ideal, namely pRp. Such rings are called local. Main article: Spectrum of a ring Spec Z contains a point for the zero ideal. The closure of this point is the entire space. The remaining points are the ones corresponding to ideals p , where p is a prime number. These points are closed.
The spectrum of a ring R, [nb 1] denoted by Spec R, is the set of all prime ideals of R. Interpreting f as a function that takes the value f mod p i. The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec R. For an algebraically closed field k, mSpec k[T1, However, the consideration of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons.
For example, the minimal prime ideals i. For a Noetherian ring R, Spec R has only finitely many irreducible components. This is a geometric restatement of primary decomposition , according to which any ideal can be decomposed as a product of finitely many primary ideals. This fact is the ultimate generalization of the decomposition into prime ideals in Dedekind rings. Affine schemes[ edit ] The notion of a spectrum is the common basis of commutative algebra and algebraic geometry.
The datum of the space and the sheaf is called an affine scheme. The resulting equivalence of the two said categories aptly reflects algebraic properties of rings in a geometrical manner.
Similar to the fact that manifolds are locally given by open subsets of Rn, affine schemes are local models for schemes , which are the object of study in algebraic geometry.
Therefore, several notions concerning commutative rings stem from geometric intuition. Main article: Krull dimension The Krull dimension or dimension dim R of a ring R measures the "size" of a ring by, roughly speaking, counting independent elements in R. The dimension is calibrated by dim k[X1, This axiom is motivated by regarding the polynomial ring in n variables as an algebraic analogue of n-dimensional space.
For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension. Similarly as for other algebraic structures, a ring homomorphism is thus a map that is compatible with the structure of the algebraic objects in question. The kernel is an ideal of R, and the image is a subring of S.
A ring homomorphism is called an isomorphism if it is bijective. An example of a ring isomorphism, known as the Chinese remainder theorem , is Z.