In particular, a commutative ring is an integral domain if and only if

The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers.

In particular, the category of finitely generated modules over a noetherian commutative ring is abelian ; in this way, abelian categories show up in commutative algebra.

Indeed, any such g lies in a particular planar commutative subring, and g has a polar coordinate form determined by parameters of radial and angular nature.

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field ( or, more generally over a commutative ring ) with an extra layer of structure, known as a gradation ( or grading ).

This property implies that any element generates a commutative associative *- algebra, so in particular the algebra is power associative.

This characterization of linear differential operators shows that they are particular mappings between modules over a commutative algebra, allowing the concept to be seen as a part of commutative algebra.

By means of a Chevalley basis for the Lie algebra, one can define E < sub > 6 </ sub > as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split ( sometimes also known as “ untwisted ”) adjoint form of E < sub > 6 </ sub >.

In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of:

However, it is true that for finitely presented modules M over a commutative ring R ( in particular if M is a finitely generated R-algebra and R is noetherian ), the following are equivalent.

In particular, parallel composition is commutative and associative.

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains.

In mathematics, a Cohen – Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality.

By means of a Chevalley basis for the Lie algebra, one can define E < sub > 8 </ sub > as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split ( sometimes also known as “ untwisted ”) form of E < sub > 8 </ sub >.

In particular, if A is commutative and x < sub > 1 </ sub > = x < sub > 2 </ sub > = … = x < sub > n </ sub >, then this formula simplifies to the familiar power rule D ( x < sup > n </ sub >) = nx < sup > n − 1 </ sup > D ( x ).

More generally it is used as a generic name for analogues for complete commutative rings ( including p-adic fields in particular ) of the Newton method for solving equations.

However there is a variation of Gauss's lemma that is valid even for polynomials over any commutative ring R, which replaces primitivity by the stronger property of co-maximality ( which is however equivalent to primitivity in the case of a Bézout domain, and in particular of a principal ideal domain ).

* Cohen – Macaulay ring, a particular type of commutative ring in mathematics

The extent of such interference -- which may be so slight as to be undetectable at any point where either of the stations renders a usable signal, or may be so great as to virtually destroy the service areas of both stations -- depends on many factors, among the principal ones being the distance between the stations, their respective radiated power, and, of particular significance here, the time of day.

The role of an earthquake in starting the destruction of whole cities is tremendously frightening, but fire may actually be the principal agent in a particular disaster.

Many of the counties of Ulster roughly correspond to the territories controlled by the principal clan in that particular area such as the O Donnells of Tír Conaill whose political power was concentrated in what would become the County of Donegal.

In computer science, a queue ( ) is a particular kind of abstract data type or collection in which the entities in the collection are kept in order and the principal ( or only ) operations on the collection are the addition of entities to the rear terminal position and removal of entities from the front terminal position.

At an umbilic all the sectional curvatures are equal ; in particular the principal curvatures are equal.

The principal parts of Leonore and Florestan, in particular, require great vocal skill and endurance in order to project the necessary intensity, and top performances in these roles attract admiration.

Frequently, a dance sequence was built around two or three principal ideas, sometimes inspired by his own steps or by the music itself, suggesting a particular mood or action.

With regard to export control laws in particular, effective enforcement is greatly enhanced by both international cooperation and an effort to harmonize the substance of U. S. laws with those of our principal trading partners.

The archaeological explorations that took place here reveal that the southern part of the city indeed contain a large number of religious buildings with a particular devotion to the god Ptah, the principal god of Memphis.

In particular, it asserts that all fractional ideals are principal, a statement which is false whenever is not a PID.

The first of these was a collection of essays on the principal truths of natural religion ( Abhandlungen von den vornehmsten Wahrheiten der natürlichen Religion, 1755, 7th ed., 1798 ); the second ( Betrachtungen über die Triebe der Thiere, 1760, 4th ed., 1798 ) dealt with one particular branch of the same subject.

The first contains a collection and exegesis of all the texts in the New Testament relating to the doctrine of the Trinity ; in the second the doctrine is set forth at large, and explained in particular and distinct propositions ; and in the third the principal passages in the liturgy of the Church of England relating to the doctrine of the Trinity are considered.

In the eparchial curia a chancellor is to be appointed who is to be a presbyter ( priest ) or deacon and whose principal obligation, unless otherwise established by the particular law, is to see that the acts of the curia are gathered and arranged as well as preserved in the archives of the eparchial curia.

In particular, the principal value of logr, where | r | = 1, can be calculated without reference to any geometrical or trigonometric construction .</ ref >

This is because the flavors supposedly reflect the names and / or principal exports of the native land: therefore, Greeks taste greasy ; Turks, taste like turkey, and are a particular favourite of Bonecruncher ; Danes taste like dogs ( Labradors-those from Labrador taste like Great Danes ), and Swedes taste sweet and sour ; people from Wales taste of fish ( because the name sounds like that of " whales "); etc.

The best player in a particular section will receive " first chair ," or the " principal seat.

The principal difference between structural genomics and traditional structural prediction is that structural genomics attempts to determine the structure of every protein encoded by the genome, rather than focusing on one particular protein.

Although the tribal shaykh was the principal figure, each Maʻdān village ( which may have contained members of several different tribes ) would also follow the authority of the hereditary qalit " headman " of a tribe's particular section.

In particular, the universal cover of X is a principal bundle over X with structure group ( since the universal cover is simply connected and thus is trivial ).

The principal sectors of intervention were the language used in the government agencies and the administration, in meetings and office spaces in particular, in corporate names, information and education.

In particular, the principal characters often discussed racial and American socio-economic class issues.

The term Grand ballabile is used if nearly all participants ( including principal characters ) of a particular scene in a full-length work perform a large-scale dance.

Technobabble's principal use in most science fiction, in particular more hard science fiction, is to conceal the true ( impossible ) nature of materials, technologies, or devices mentioned in the story, frequently because of a violation regarding the current understanding regarding the laws of physics.

* There is a free and transitive group action of G on the collection of frames: it is a principal homogeneous space for G. In particular, for any pair of frames ƒ and ƒ ′, there is a unique transition of frame ( ƒ → ƒ ′) in G determined by the requirement ( ƒ → ƒ ′) ƒ = ƒ ′.

A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.