The notion of a graded module is the generalization of graded vector spaces.

In particular he invented the notion of vertex algebras, which Igor Frenkel, James Lepowsky and Arne Meurman used to construct an infinite-dimensional graded algebra acted on by the monster group.

Explicitly, it is used to define the degree of a polynomial and the notion of homogeneous polynomial, as well as for graded monomial orderings used in formulating and computing Gröbner bases.

This leads to a graded notion of categories, which is a central notion in many models of cognitive science and cognitive semantics, e. g. in the work of George Lakoff ( Women, Fire and Dangerous Things, 1987 ) or

family resemblance is calling for a notion of conceptual distance, which is closely related to the idea of graded sets, but there are problems as well.

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.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring.

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

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 ".

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.

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.

His notion of abelian category is now the basic object of study in homological algebra.

In view of implacable demands by the political leadership of both Hindu and Islamic communities in British India for a separate Hindu and a separate Muslim homeland, Mountbatten conceded the notion of two nations consisting of a Hindu-majority India and a Muslim-majority Pakistan ( which incorporated East Pakistan, now Bangladesh ).

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874 – 1884.

In it he stated another notion of " effective computability " with the introduction of his a-machines ( now known as the Turing machine abstract computational model ).

Parallel with this generational change, the articles and books on the doctrine and practice of fellowship with the main " Central " grouping now reject the notion itself of separate " fellowships " among those who recognise the same baptism as " schism ".

The notion of temporary insanity argues that a defendant was insane, but is now sane.

The actual titles are in question ( as is the notion that these were album-length recordings ), but the first title has been confirmed by a former head of the now defunct Smelly Tongues fan club.

But now another problem about the notion that God is a mind arises.

Adams, once a political visionary and now a man blind to the needs of his country, not only held fast to his notion of defending the monarchy but also made additional attempts to form similarly flawed Federation-like entities after that union's demise.

In 1966 Per Martin-Löf introduced a new notion which is now generally considered the most satisfactory notion of algorithmic randomness.

The fact that many immigrants settle in impoverished city centres led to the notion of the " peripheralisation of the core ", which simply describes that people who used to be at the periphery of the former empires now live right in the centre.

Though transformations continue to be important in Chomsky's current theories, he has now abandoned the original notion of Deep Structure and Surface Structure.

Scholars now reject the former notion, but note that there were certain traditions and practices used in both the Irish and British churches but not in the wider Christian world.

Nagel and Newman can now define the notion of tautologous: " a formula is a tautology if, and only if, it falls in the class K < sub > 1 </ sub > no matter in which of the two classes its elements are placed ".

To Be or Not To Be, now regarded as one of the best films of Lubitsch's, Benny's and Lombard's careers, was initially not well received by the public, many of whom could not understand the notion of making fun out of such a real threat as the Nazis.

Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path ( depending on the type of differential structure applied ).

This essay, first published in the Collected Works of 1894-1911, contains the first known appearance of what are now called linear algebra and the notion of a vector space.

The notion of a personal life, as now understood, is in part a creation of modern Western society.

However, this usage is now uncommon, as the notion of " just plain ASCII " ( though very attractive at first glance ), has turned out to have difficulties.

The notion of local hidden variables is now introduced by considering the following question:

It can now run atop TCP with a small adaptation layer that adds a packet length to each SMB message ; this is necessary because TCP only provides a byte-stream service with no notion of packet boundaries.

The U. S. government has tried to sell the notion that the war against Iraq is critical to the American " War on Terrorism ": " In the war on terror, Iraq is now the central front ..." President Bush said on December 14, 2005 ( Bush, George W. " President Discusses Iraqi Elections, Victory in the War on Terror.

In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory.