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Concept
Commutative ring
Summary
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by "" and ""; e.g. and . To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., . The identity elements for addition and multiplication are denoted and , respectively.
If the multiplication is commutative, i.e.
then the ring is called commutative. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.
An important example, and in some sense crucial, is the ring of integers with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted as an abbreviation of the German word Zahlen (numbers).
A field is a commutative ring where and every non-zero element is invertible; i.e., has a multiplicative inverse such that . Therefore, by definition, any field is a commutative ring. The rational, real and complex numbers form fields.
If is a given commutative ring, then the set of all polynomials in the variable whose coefficients are in forms the polynomial ring, denoted . The same holds true for several variables.
If is some topological space, for example a subset of some , real- or complex-valued continuous functions on form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for a complex manifold.
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Inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any although their existence depends on the category that is considered. They are a special case of the concept of in category theory. By working in the , that is by reverting the arrows, an inverse limit becomes a direct limit or inductive limit, and a limit becomes a colimit.
Principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.
Spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring R is the set of all prime ideals of R, and is usually denoted by ; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings . For any ideal I of R, define to be the set of prime ideals containing I. We can put a topology on by defining the to be This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For f ∈ R, define Df to be the set of prime ideals of R not containing f.