Definitions

1. Integral domains are sometimes allowed to be noncommutative, but we adopt the convention that an integral domain is commutative unless otherwise specified.
2. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. (Web site)
3. Integral domains are also characterized by the condition that they are reduced and irreducible. (Web site)
4. Fields and integral domains are very important in modern algebra. (Web site)

Category

1. The category of fields is a reflective subcategory of the category of integral domains.

Particular

1. In particular, the only finite integral domains are the finite fields. (Web site)

Euclidean Domains

1. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. (Web site)

Injective

1. Let C be the category of integral domains and injective ring maps. (Web site)

Pid

1. It turns out that there is a subclass of all integral domains in which the notions of UFD and PID are equivalent. (Web site)

Polynomials

1. This is certainly not true of ring homomorphisms, even integral domains, as the polynomials over Z map to Z, simply by extracting the constant term.
2. Integral domains; ordered fields; rational, real, and complex numbers; polynomials and theory of equations. (Web site)

Factorization

1. Commutative algebra, finite abelian groups, combinatorics and number theory: especially factorization of elements in integral domains and monoids.

First Definition

1. Prime rings, under the first definition, can be regarded as a simultaneous generalization of both integral domains and matrix rings over fields. (Web site)

Polynomial Rings

1. Polynomial rings, factorization in R [X] and in integral domains.

Maximal Ideals

1. Definition and properties of integral domains, Fields, Prime and maximal ideals, Fields of quotients.

Fields

1. An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. (Web site)
2. Integral domains and fields are rings and the above definitions for isomorphisms and homomorphisms also apply to integral domains and fields.
3. A package of "generic" Maple routines for doing linear algebra over arbitrary commutative rings, integral domains, Euclidean domains and fields.

Division Rings

1. In particular, this applies to all fields, to all integral domains, and to all division rings. (Web site)

Irreducible

1. It follows that the unique minimal ideal of a reduced and irreducible ring is the zero ideal, hence such rings are integral domains. (Web site)

Integral Domain

1. Rings of polynomials are integral domains if the coefficients come from an integral domain.
2. Rings of polynomial s are integral domains if the coefficients come from an integral domain. (Web site)

Free Field

1. For example, neither the free field over any set nor the direct product of integral domains exists.

Integral Domains

1. Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subring s of fields. (Web site)
2. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. (Web site)
3. These include the full subcategories of commutative rings, integral domains, principal ideal domains, and fields.

Categories

1. Physics > Quantum Theory > Quantum Field Theory > Free Field
2. Integral Domain
3. Division Rings
4. Commutative Rings
5. Polynomial Rings
6. Books about "Integral Domains" in Amazon.com
   

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