In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus every module direct summand of R is generated by an idempotent.

Thus, if the pitch diameter of a gear is 40 mm and the number of teeth 20, the module is 2, which means that there are 2 mm of pitch diameter for each tooth.

Thus a finitely generated torsionfree module of rank can be expressed as, where is a rank one projective module.

Given two tensor bundles E → M and F → M, a map A: Γ ( E ) → Γ ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A ( fs ,...) = fA ( s ,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C < sup >∞</ sup >( M )- linear map on the module of sections.

Thus, a document retrieval module uses search engines to identify the documents or paragraphs in the document set that are likely to contain the answer.

Thus a projective module is locally free.

Suppose that R is an algebra over the field C of complex numbers and M = N is a finite-dimensional simple module over R. Then Schur's lemma says that the endomorphism ring of the module M is a division ring ; this division ring contains C in its center, is finite-dimensional over C and is therefore equal to C. Thus the endomorphism ring of the module M is " as small as possible ".

Thus, it is a measure of how strongly-related each piece of functionality expressed by the source code of a software module is.

Thus here only one module can raise the IRQ signal at a time.

Thus module can be extended to mean more generally a unitary part that gives the measuring unit for the whole.

Thus, the extent to which language processing occurs within an anatomical module is considered to be minimal.

Thus, functional module status cannot be granted to language processing based on this evidence.

Thus, the evidence from double dissociations does not support modularity, although it should be noted that lack of dissociation is not evidence against a module ; this inference cannot be logically made.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus ideas like `` grace '', `` salvation '', and `` providence '' cluster together in traditional Christianity.

Thus numbers like 2 and 120 ( 2 × 60 ), 3 and 180 ( 3 × 60 ), 4 and 240 ( 4 × 60 ), looked the same because the larger numbers lacked a final sexagesimal placeholder.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Thus one finds brass instruments made of wood, like the alphorn, the cornett, the serpent and the didgeridoo, while some woodwind instruments are made of brass, like the saxophone.

Thus, one might pursue an increase in material equality or political liberty instead of something like the more ephemeral " pleasure ".

Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons.

Thus, texts like the Annals will often state whether the calendar they use ( the calendar of Lu ) is in phase with the Royal calendar ( used by the Zhou kings ).

) Thus the alveolar click sounds something like a cork pulled from a bottle ( a low-pitch pop ), at least in Xhosa ; while the dental click is like English tsk!

Thus, the Debian Project is an independent decentralized organization ; it is not backed by a company like Linux distributions such as Ubuntu, openSUSE, Fedora, and Mandriva.

" Thus Thrax, like contemporary Alexandrian scholars who edited Attic Greek and Homeric texts, was concerned with facilitating the teaching of classic Greek literature to an audience who spoke Koine Greek.

Thus, one might pursue an increase in material equality or political liberty instead of something like the more ephemeral " pleasure ".

Thus, during the 1960s and early 1970s, Portuguese development plans promoting strong economic growth and effective socioeconomic policies, like those applied by the Portuguese in the other two theaters of war ( Portuguese Angola and Portuguese Mozambique ), were not possible.

Thus like many of Cassidy's derivations, this proposed explanation appears quite improbable.

Thus we have vertical datums like the NAP ( Normaal Amsterdams Peil ), the North American Vertical Datum 1988 ( NAVD88 ), the Kronstadt datum, the Trieste datum, and so on.

Thus, many typical household applications already in existence like word processing and email were designed specifically to avoid using floating point operations.

Thus the liberty of the journalist was to be dedicated to gathering verifiable facts while commentators like himself would place the news in the broader perspective.

Thus an example of the expression would look like this:

Thus, this work of Vyasa, called Jaya deals with diverse subjects like geography, history, warfare, religion and morality.

Thus, like the 68008, it could be used in systems with cheaper 8-bit memories.

" Thus it serves as a reminder to be humble, and to not forget what life was like in servitude.

Thus they are more like a two-way radio than like a telephone.

Thus, the vector space B ( X, Y ) can be given the operator norm

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

Thus, the divergence of the vector field F can be expressed as:

Thus, the Jones vector represents ( relative ) amplitude and ( relative ) phase of electric field in x and y directions.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus the area of S < sub > D </ sub > is obtained by integrating the length of the normal vector to the surface over the appropriate region D in the parametric uv plane.

Thus force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.

Thus the spinor rotation γ ( ψ ) = γψ ( ordinary quaternionic multiplication ) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation.

Thus, it is easy to convert from a vector file to a range of bitmap / raster file formats but it is much more difficult to go in the opposite direction, especially if subsequent editing of the vector picture is required.

Thus two arrows and in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB ′ A ′ is a parallelogram.

Thus the bound vector represented by ( 1, 0, 0 ) is a vector of unit length pointing from the origin along the positive x-axis.

Thus, this equation is sometimes known as the vector wave equation.

Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.

Thus if γ ( s ) is the arclength parametrization of C then the unit tangent vector T ( s ) is given by

Thus, suppose that ( x < sub > 1 </ sub >,..., x < sub > n </ sub >) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are

Thus the act of measurement induces a fundamental irreversible change on the density matrix ; this is analogous to the " collapse " of the state vector, or wavefunction collapse.

Thus, acceleration is the second derivative of the position vector that defines the trajectory of a particle.

Thus, the magnitude of the angular velocity vector at a given time t is consistent with the two dimensions case.

Thus Tychonoff's theorem joins several other basic theorems ( e. g. that every nonzero vector space has a basis ) in being equivalent to AC.

Thus, G < sub > t </ sub >( V ) = exp ( tV ) is the exponential map of the vector tV.

Thus, for every vector

Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z-axis.