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. In such a situation S is also called an R -algebra, by understanding that s in S may be multiplied by some r of R , by setting.
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. Commutative rings, together with ring homomorphisms, form a category. By means of this map, an integer n can be regarded as an element of R. For example, the binomial formula. Given two R -algebras S and T , their tensor product.
In some cases, the tensor product can serve to find a T -algebra which relates to Z as S relates to R. For example,. An R -algebra S is called finitely generated as an algebra if there are finitely many elements s 1 , Equivalently, S is isomorphic to.
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A much stronger condition is that S is finitely generated as an R -module , which means that any s can be expressed as a R -linear combination of some finite set s 1 , A ring is called local if it has only a single maximal ideal, denoted by m. For any not necessarily local ring R , the localization. This localization reflects the geometric properties of Spec R "around p ".
Several notions and problems in commutative algebra can be reduced to the case when R is local, making local rings a particularly deeply studied class of rings. The residue field of R is defined as. Informally, the elements of m can be thought of as functions which vanish at the point p , whereas m 2 contains the ones which vanish with order at least 2. For any Noetherian local ring R , the inequality. If equality holds true in this estimate, R is called a regular local ring. A Noetherian local ring is regular if and only if the ring which is the ring of functions on the tangent cone.
Broadly speaking, regular local rings are somewhat similar to polynomial rings. Discrete valuation rings are equipped with a function which assign an integer to any element r. This number, called the valuation of r can be informally thought of as a zero or pole order of r. Discrete valuation rings are precisely the one-dimensional regular local rings.
For example, the ring of germs of holomorphic functions on a Riemann surface is a discrete valuation ring. By Krull's principal ideal theorem , a foundational result in the dimension theory of rings , the dimension of. A ring R is called a complete intersection ring if it can be presented in a way that attains this minimal bound. This notion is also mostly studied for local rings. Any regular local ring is a complete intersection ring, but not conversely.
A ring R is a set-theoretic complete intersection if the reduced ring associated to R , i. As of , it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections. The depth of a local ring R is the number of elements in some or, as can be shown, any maximal regular sequence, i. A local ring in which equality takes place is called a Cohen—Macaulay ring.
Local complete intersection rings, and a fortiori, regular local rings are Cohen—Macaulay, but not conversely. Cohen—Macaulay combine desirable properties of regular rings such as the property of being universally catenary rings , which means that the co dimension of primes is well-behaved , but are also more robust under taking quotients than regular local rings. There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable.
For example, an integral domain that is integrally closed in its field of fractions is called normal. This is a desirable property, for example any normal one-dimensional ring is necessarily regular. Rendering [ clarification needed ] a ring normal is known as normalization.
If I is an ideal in a commutative ring R , the powers of I form topological neighborhoods of 0 which allow R to be viewed as a topological ring.
This topology is called the I -adic topology. R can then be completed with respect to this topology. For example, if k is a field, k [[ X ]], the formal power series ring in one variable over k , is the I -adic completion of k [ X ] where I is the principal ideal generated by X.
This ring serves as an algebraic analogue of the disk. Analogously, the ring of p -adic integers is the completion of Z with respect to the principal ideal p. Any ring that is isomorphic to its own completion, is called complete. Complete local rings satisfy Hensel's lemma , which roughly speaking allows extending solutions of various problems over the residue field k to R.
Several deeper aspects of commutative rings have been studied using methods from homological algebra.
Hochster lists some open questions in this area of active research. Projective modules can be defined to be the direct summands of free modules.
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If R is local, any finitely generated projective module is actually free, which gives content to an analogy between projective modules and vector bundles. A local Noetherian ring is regular if and only if its global dimension is finite, say n , which means that any finitely generated R -module has a resolution by projective modules of length at most n. The proof of this and other related statements relies on the usage of homological methods, such as the Ext functor.
This functor is the derived functor of the functor. The higher Ext functors measure the non-exactness of the Hom-functor. The importance of this standard construction in homological algebra stems can be seen from the fact that a local Noetherian ring R with residue field k is regular if and only if. Moreover, the dimensions of these Ext-groups, known as Betti numbers , grow polynomially in n if and only if R is a local complete intersection ring.
The tensor product is another non-exact functor relevant in the context of commutative rings: for a general R -module M , the functor. If it is exact, M is called flat. Despite being defined in terms of homological algebra, flatness has profound geometric implications. For example, if an R -algebra S is flat, the dimensions of the fibers.
By Wedderburn's theorem , every finite division ring is commutative, and therefore a finite field. More general conditions which guarantee commutativity of a ring are also known. R is called a commutative differential graded algebra cdga. An example is the complex of differential forms on a manifold , with the multiplication given by the exterior product , is a cdga. The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. A broad range examples of graded rings arises in this way.
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For example, the Lazard ring is the ring of cobordism classes of complex manifolds. A related notion is an almost commutative ring , which means that R is filtered in such a way that the associated graded ring. An example is the Weyl algebra and more general rings of differential operators. A simplicial commutative ring is a simplicial object in the category of commutative rings. They are building blocks for connective derived algebraic geometry.
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The ring of matrices is not commutative, since matrix multiplication fails to be commutative. This fact makes representations of commutative Lie groups particularly simpler to understand than in general. An example is the set of matrices of divided differences with respect to a fixed set of nodes. From Wikipedia, the free encyclopedia. Algebraic structures Group -like.
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