This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

For this reason, the notion of a Noetherian ring generalizes such rings as.

Thus, in a sense, the notion of a Noetherian ring unifies the ideal structure of various " natural rings ".

Global dimension is an important technical notion in the dimension theory of Noetherian rings.

This notion of ideal coincides with the notion of ring ideal in the Boolean ring A.

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.

In ring theory, the notion of number is generally replaced with that of ideal.

Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring.

The concept of an order ideal in order theory is derived from the notion of ideal in ring theory.

In fact, in the general case of a ring, these two requirements define the notion of an ideal in a ring.

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions which successfully extend the notion to rings without unity.

When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.

A ring already has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.

The notion of " graded ring " now becomes the same thing as a N-graded ring, where N is the monoid of non-negative integers under addition.

Nevertheless, there is still a notion of kernel studied in ring theory.

The notion of root of unity also applies to any algebraic ring with a multiplicative identity element, namely a root of unity is any element of finite multiplicative order.

One may generalize this notion by replacing the field of scalars by a commutative ring, and thus defining an algebra over a ring.

There is a haunting resemblance between the notion of cause in Copernicus and in Freud.

Within this notion clarity is possible, but for us who are neither Greek nor Jansenist there is not such clarity.

Hence, the only defensible procedure is to repress any and every notion, unless it gives evidence that it is perfectly safe.

This is the principal point made in this final section of Englishman No. 57, and it caps Steele's efforts in his other writing of these months to counteract the notion of the Tories as a `` Church Party '' supported by the body of the clergy.

The notion of `` inspiration '' is somehow cognate to this feeling.

The Aristotelian notion of catharsis, the purging of emotion, is a persistent and viable one.

The idea here is one of discharge but this must stand in opposition to a second view, Plato's notion of the arousal of emotion.

This is given some expression in Beardsley's notion of harmony and the resolution of indecision.

And there is one other point in the Poetics that invites moral evaluation: Aristotle's notion that the distinctive function of tragedy is to purge one's emotions by arousing pity and fear.

I refer to the notion that the structure of society is a microcosm of the cosmic design and that history conforms to patterns of justice and chastisement as if it were a morality play set in motion by the gods for our instruction.

That notion is fantastically wrong-headed from several points of view.

The notion of philosophy as Queen Bee may fit well with authoritarian modes of political ideology, but it has been noted that the price of such an imperial notion of philosophy is the frustration and flagellation of the social sciences.

The threadbare notion that belief, unlike behaviour, is not subject to objective analysis, has placed intuitive metaphysics squarely against the sociology of knowledge, since it is precisely the job of the sociology of knowledge to treat beliefs as social facts no less viable than social behaviour.

These cases, for all their rarity, are so dramatic that friends and relations repeat the story until the general population may get an entirely false notion of how often the hymen is a serious problem to newly-weds.

It is a notion which contains a gratuitous insult, implying, as it does, that Negroes can make no move unless they are manipulated.

They would like to convey the notion something is being done, even though it is something they know to be ineffectual.

( Pp. 228-229 ) in any event, it is obvious that the anti-trust laws did not prevent the formation of some of the greatest financial empires the world has ever known, held together by some of the most fantastic ideas, all based on the fundamental notion that a corporation is an individual who can trade and exchange goods without control by the government ''.

Still, the notion of altruism is modified in such a world-view, since the belief is that such a practice promotes our own happiness: " The more we care for the happiness of others, the greater our own sense of well-being becomes " ( Dalai Lama ).

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis.

Even though the three processes mentioned above proved to be equivalent, the fundamental premise behind the thesis — the notion of what it means for a function to be effectively calculable — is " a somewhat vague intuitive one ".

The Albigensians, more commonly known as the Cathars, were a heretical gnostic sect, holding that matter was evil and only spirit was good ; this was a fundamental challenge to the notion of incarnation, central to Roman Catholic theology.

A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of " discharging " developed by Heesch.

Groups share a fundamental kinship with the notion of symmetry.

The debate about this notion as regards physicalism is to what extent mental phenomena exist independently of their ( supposed ) fundamental lowest level-the physical.

Influential to thinkers associated with Postmodernism are Heidegger's critique of the subject-object or sense-knowledge division implicit in Rationalism, Empiricism and Methodological Naturalism, his repudiation of the idea that facts exist outside or separately from the process of thinking and speaking them ( however, Heidegger is not specifically a Nominalist ), his related admission that the possibilities of philosophical and scientific discourse are wrapped up in the practices and expectations of a society and that concepts and fundamental constructs are the expression of a lived, historical exercise rather than simple derivations of external, apriori conditions independent from historical mind and changing experience ( see Johann Gottlieb Fichte, Heinrich von Kleist, Weltanschauung and Social Constructionism ), and his Instrumentalist and Negativist notion that Being ( and, by extension, reality ) is an action, method, tendency, possibility and question rather than a discreet, positive, identifiable state, answer or entity ( see also Process Philosophy, Dynamism, Instrumentalism, Pragmatism and Vitalism ).

In axiomatic set theory the fundamental concept of set is an example of a primitive notion.

IV, p. 164 ): " The fundamental tenet of the Vedânta school consisted not in denying the existence of matter, that is solidity, impenetrability, and extended figure ( to deny which would be lunacy ), but in correcting the popular notion of it, and in contending that it has no essence independent of mental perception ; that existence and perceptibility are convertible terms.

In either approach, the fundamental notion is that of an isotropic subspace W. Each construction depends on an initial freedom in choosing this subspace.

The notion of the fundamental role of non-cognitive understanding in rendering possible thematic consciousness informed the discussion surrounding Artificial Intelligence during the 1970s and 1980s.

John Locke's conception of the social contract differed from Hobbes ' in several fundamental ways, retaining only the central notion that persons in a state of nature would willingly come together to form a state.

It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of " meaning " itself.

The notion of an " incorporeal potency " is similar to that of " action at a distance " except the former notion remained purely qualitative and there is no inkling of the fundamental " action and reaction " principle.

More profoundly, Shakespeare became a source, by way of its dramatic truth, for Berlioz ' fundamental notion of expressive truth ; this was how he could call Romeo and Juliet " the supreme drama of my life.

In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.

Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities.

Those who argue classism is especially pervasive or fundamental to the society that they live in often identify classism as the expression of systematic economic exploitation by the ' higher ' classes, and may connect it with an explicit notion of class warfare.

All proofs below use the notion of the adjugate matrix of an n × n matrix M. This is a matrix whose coefficients are given by polynomial expressions in the coefficients of M ( in fact by certain ( n − 1 )×( n − 1 ) determinants ), in such a way that one has the following fundamental relations

In response, it has been argued that the notion of the Duplessis " black years " is a myth propagated by all subsequent major political actors in Quebec due to a fundamental aversion to Catholic church-oriented traditionalist patterns of development, with dominant intellectual movements combining various elements of this dislike.

However, they display certain features that make them fundamental in exploring the notion of diagonalizability of certain operators on vector spaces.

The code that performs hash, compare, or extend operations must also be protected — in this context, the notion of an immutable root-of-trust holds that the very first code to measure security properties of a system must itself be trusted to ensure that a rootkit or bootkit does not compromise the system at its most fundamental level.