If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that ( a ) I = ( b ) J.

If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets.

Humphrey was an enthusiastic successor of his father's New Deal-inspired political philosophy, and throughout his career he remained devoted to traditional progressive ideals as well as their more modern manifestations: " If you think that being too liberal means raising the minimum wage, advocating health care for everyone, protecting the environment, taking on the tobacco industry, enacting campaign finance reform, and putting more cops on the streets, then guess what?

If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I.

If we admit that the concretization of ideals genuinely occurs, Royce argues, then we are not only entitled but compelled to take seriously and regard as real the larger intelligible structures within which those ideals exist, which is the purposive character of the divine Will.

If America is failing to reach them today, it's not because our ideals need replacing, it's because our President needs replacing.

* If I, J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form.

* If a given Lie algebra g is a direct sum of its ideals I < sub > 1 </ sub >,..., I < sub > n </ sub >, then the Killing form of g is the direct sum of the Killing forms of the individual summands.

If a number does not factor uniquely into primes, then the ideal generated by the number may still factor into the intersection of powers of prime ideals.

If he was not ' prepared ... to continue in the style, and with the ideals ' of the diocese, then he was to ' part company ' with the Diocese.

If is a closed immersion and is the quasi-coherent sheaf of ideals cutting out Z, then the direct image from the category of quasi-coherent sheaves over Z to the category of quasi-coherent sheaves over X is exact, fully faithful with the essential image consisting of such that.

If not already sympathetic, he was probably won over to Luther's ideals at this meeting.

If and are prime ideals of A and B, respectively, such that

Put another way, a sieve is a collection S of arrows with a common codomain which satisfies the functoriality condition, " If g: c ′→ c is an arrow in S, and if f: c ″→ c ′ is any other arrow in C, then the pullback is in S ." Consequently sieves are similar to right ideals in ring theory or filters in order theory.

" If we do not join forces with Muslim dissident and feminist groups ; and, above all, if we do not have one universal standard of human rights for all — then we will fail our own Judeo-Christian and secular Western ideals.

If Δ denotes the relative discriminant of L / K, the Artin symbol ( or Artin map, or ( global ) reciprocity map ) of L / K is defined on the group of prime-to-Δ fractional ideals,, by linearity:

If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

If the valid element indices begin at 0, the constant B is simply the address of the first element of the array.

If the numbering does not start at 0, the constant B may not be the address of any element.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If and we have for all v, w in V, then we say that B is symmetric.

In addition to acting and occasionally directing, Campbell has become a writer, starting with an autobiography, If Chins Could Kill: Confessions of a B Movie Actor published on August 24, 2002.

If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

* Indicative conditional, a conditional in the form of " If A then B " in natural languages

If an atom A is double-bonded to an atom B, A is treated as being singly bonded to two atoms: B and a ghost atom that has the same atomic number as B but is not attached to anything except A.

If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.

If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

If Af denotes the net profit from stage R and Af, then the principle of optimality gives Af.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

* If the balance factor of P is-2 then the right subtree outweighs the left subtree of the given node, and the balance factor of the right child ( R ) must be checked.

* If the balance factor of R is-1, a single left rotation ( with P as the root ) is needed ( Right-Right case ).

* If the balance factor of R is + 1, two different rotations are needed.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

If the thumb points in the direction of the 4th substitutent, the enantiomer is R. Otherwise, it's S.

If the relative priorities of these substituents need to be established, R takes priority over S. When this happens, the descriptor of the stereocenter is a lowercase letter ( r or s ) instead of the uppercase letter normally used.

If a is a point in R < sup > n </ sup >, then the higher dimensional chain rule says that:

If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

If, i. e., it has a large norm with each value of s, and if, then Y ( s ) is approximately equal to R ( s ) and the output closely tracks the reference input.

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

* If x < sub > 0 </ sub > is a real number, we can turn the set R −

( If X is also empty then R is reflexive.

*( EF1 ) If a and b are in R and b is nonzero, then there are q and r in R such that and either r = 0 or.

If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.