* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given a sample of wood, the variation of the tree ring growths provides not only a match by year, it can also match location because the climate across a continent is not consistent.

Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:

Given a Boolean ring R, for x and y in R we can define

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

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.

; Factor ring or quotient ring: Given a ring R and an ideal I of R, the factor ring is the ring formed by the set R / I of cosets

Given a ring R and a subset S, one wants to construct some ring R * and ring homomorphism from R to R *, such that the image of S consists of units ( invertible elements ) in R *.

Given a *- ring, there is also the map.

Given a module M over a ring R, an R endomorphism f of M is called an involution if f < sup > 2 </ sup > is the identity homomorphism on M.

Given a module A over a ring R, and a submodule B of A, the quotient space A / B is defined by the equivalence relation

Given an integral domain, let be an element of, the polynomial ring with coefficients in.

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

: Given: a function f: A R from some set A to the real numbers

# Given u in W and a scalar c in R, if u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) again, then cu = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0 ) = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function:

Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module.

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given an evaluation e of variables by elements of M < sub > w </ sub >, we

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Given an SVD of M, as described above, the following two relations hold:

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

Given high enough cooling rates and appropriate alloy composition, metallic bonding can occur even in glasses with an amorphous structure.

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

Given the composition of the current 41st Canadian Parliament, once the Official Opposition, the New Democratic Party, is finished, the next question comes from the only other officially-recognized opposition party, the Liberal Party.

Given that socks directly contact the feet, their composition can have an impact on foot odor.

Given two subsets A and B of N and a set of functions F from N to N which is closed under composition, A is called reducible to B under F if

Position Statement Summary: Given that the focus of writing instruction is expanding: the curriculum of composition is widening to include not one but two literacies: a literacy of print and a literacy of the screen and that work in one medium is used to enhance learning in the other, this statement articulates principles of good practice governing these new pedagogical activities for digital literacy.

Given a map, the mapping cylinder is a space, together with a cofibration and a surjective homotopy equivalence ( indeed, Y is a deformation retract of ), such that the composition equals f.

Given the widely recognized importance of proteins as cellular catalysts and recognition elements, the ability to precisely control the composition and connectivity of polypeptides is a valued tool in the chemical biology community and is an area of active research.

Given the timing of Led Zeppelin's first live performances of " Since I've Been Loving You " ( early 1970 ), it has been suggested that " Never " is the original source for this composition.

Additionally, there are composition operations: Given a sequence of sequences of objects, a sequence of objects, and an object Z: if

Given its unusual composition for an igneous rock, the magmatic nature of the carbonatite was not proposed for a long time and remained doubtful subsequently.