constant and this relating the variables P. V and T of an ideal gas is known as the equation of state.

Define, where is the discrete valuation corresponding to the ideal P < sub > i </ sub >.

Since the ideal windings have no impedance, they have no associated voltage drop, and so the voltages V < sub > P </ sub > and V < sub > S </ sub > measured at the terminals of the transformer, are equal to the corresponding EMFs.

The set P here consists of all ( two-sided ) ideals in R except R itself, which is not empty since it contains at least the trivial ideal

This is sometimes written as V = k T, where k is a constant dependent on the type, mass, and pressure of the gas and T is temperature on an absolute scale ( in terms of the ideal gas law, k = n · R / P ).

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

where a is the radius of the particle A, M < sub > A </ sub > is the molecular mass of the particle A, D < sub > AB </ sub > is the diffusion coefficient between particles A and B, R is the ideal gas constant, T is the temperature in kelvins and P are the pressures at infinite and at the surface respectively.

where P is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas ( also known as number of moles ), T is the temperature of the gas and R is the ideal, or universal, gas constant, equal to the product of Boltzmann's constant and Avogadro's constant.

Wallace also had guest appearances on Charles In Charge, Murder, She Wrote, Magnum, P. I., and A Different World, On Taxi, she portrayed herself, chosen as the ideal date of Rev.

Unlike a classical ideal gas, whose pressure is proportional to its temperature ( P = nkT / V, where P is pressure, V is the volume, n is the number of particles — typically atoms or molecules — k is Boltzmann's constant, and T is temperature ), the pressure exerted by degenerate matter depends only weakly on its temperature.

A Sloan Foundation grant established the MIT School of Industrial Management in 1952 with the charge of educating the " ideal manager ", and the school was renamed in Sloan's honor as the Alfred P. Sloan School of Management, one of the world's premier business schools.

For ideal gases the bulk modulus P is simply the gas pressure multiplied by the adiabatic index.

An Alfred P. Sloan Foundation grant established the MIT School of Industrial Management in 1952 with the charge of educating the " ideal manager ", and the school was renamed in Sloan's honor as the Alfred P. Sloan School of Management.

Simply, any prime ideal P has a corresponding residue field, which is the field of fractions of the quotient A / P, and any element of A has a reflection in this residue field.

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

The dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters, thus, for an ideal gas described with three thermodynamic parameters P, V and n, it is a two-dimensional surface.

For example, if two systems of ideal gases are in equilibrium, then P < sub > 1 </ sub > V < sub > 1 </ sub >/ N < sub > 1 </ sub > = P < sub > 2 </ sub > V < sub > 2 </ sub >/ N < sub > 2 </ sub > where P < sub > i </ sub > is the pressure in the ith system, V < sub > i </ sub > is the volume, and N < sub > i </ sub > is the amount ( in moles, or simply the number of atoms ) of gas.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors ( e. g., Bourbaki ) refer to PIDs as principal rings.

* Nilradical of a ring, the nilradical of a commutative ring is a nilpotent ideal, which is as large as possible

When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

* If R is a unital commutative ring with an ideal m, then k = R / m is a field if and only if m is a maximal ideal.

* In a commutative ring with unity, every maximal ideal is a prime ideal.

* In commutative algebra, a commutative ring can be completed at an ideal ( in the topology defined by the powers of the ideal ).

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.

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

** Every unital ring other than the trivial ring contains a maximal ideal.

* Generating set of an ideal ( ring theory ):

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

As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain ( PID ).

* Conductor ( ring theory ), an ideal of a ring that measures how far it is from being integrally closed

The property ( EF1 ) can be restated as follows: for any principal ideal I of R with nonzero generator b, all nonzero classes of the quotient ring R / I have a representative r with.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra + Rb is principal and indeed is equal to Rd.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

The main concept, however, common to all idealist epistemologies is the centrality of Reason: ( i. e.: ' Reason ' with a capital ' R '): a priori Reason: Knowledge can only be, ultimately, a product of the mind and is therefore, by definition, ' ideal '.

* R is a principal ideal domain.

In fact, if I is a nonzero ideal of R then any element a of I

Universalist theories are generally forms of moral realism, though exceptions exists, such as the subjectivist ideal observer and divine command theories, and the non-cognitivist universal prescriptivism of R. M.

For instance, the ring ( in fact field ) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial.

: R is the ideal gas constant.

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.