Let-R-be-a-ring-and-G-be-a-monoid-.

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Let and R

Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

Let R be a fixed commutative ring.

Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.

The Beatles ' 1968 track " Back in the U. S. S. R " references the instrument in its final verse (" Let me hear your balalaikas ringing out / Come and keep your comrade warm ").

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Let R denote the field of real numbers.

Let R be an integral domain.

Let R be a domain and f a Euclidean function on R. Then:

Gloria Gaynor ( born September 7, 1949 ) is an American singer, best known for the disco era hits ; " I Will Survive " ( Hot 100 number 1, 1979 ), " Never Can Say Goodbye " ( Hot 100 number 9, 1974 ), " Let Me Know ( I Have a Right )" ( Hot 100 number 42, 1980 ) and " I Am What I Am " ( R & B number 82, 1983 ).

Let us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable.

Let R be the quadratic mean ( or root mean square ).

* Let R :=

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ x , defined by

Let R 2n have the basis

If V is a real vector space, then we replace V by its complexification V ⊗ R C and let g denote the induced bilinear form on V ⊗ R C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g | W = 0.

Let V be a vector space over a field K, and let be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field.

Let R be the set of all sets that are not members of themselves.

Let ( M, g ) be a Riemannian manifold and ƒ: M m → R n a short C ∞- embedding ( or immersion ) into Euclidean space R n , where n ≥ m + 1.

Let U and V be two open sets in R n .

Let and be

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.

`` Let him be now ''!!

Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

Let this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??

Let and ring

One means to help the birds occurs to me: Let the chimes that ring over Washington Square twice daily, discontinue any piece of music but one.

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.

* Let A be a commutative ring with unity and let S be a multiplicative subset of A.

Let M and N be ( left or right ) modules over the same ring, and let f: M → N be a module homomorphism.

Let denote the ring of integers ; that is, let be the set of integers equipped with its natural operations of addition and multiplication.

Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:

* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I n be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I 0 , I 1 , I 2 , etc., is an ascending chain that does not terminate.

: Let freedom ring!

Let k be a field ( such as the rational numbers ) and K be an algebraically closed field extension ( such as the complex numbers ), consider the polynomial ring kX n and let I be an ideal in this ring.

Let be a Jacobson ring.

Let freedom ring.

Let F be a field and p ( X ) be a polynomial in the polynomial ring F of degree n. The general process for constructing K, the splitting field of p ( X ) over F, is to construct a sequence of fields such that is an extension of containing a new root of p ( X ).

Let k be an algebraically closed field and let A n be an affine n-space over k. The polynomials ƒ in the ring k ..., x n can be viewed as k-valued functions on A n by evaluating ƒ at the points in A n , i. e. by choosing values in k for each x i .

Let k be the field of complex numbers C. Let A 2 be a two dimensional affine space over C. The polynomials f in the ring Cy can be viewed as complex valued functions on A 2 by evaluating ƒ at the points in A 2 .

Let again k be the field of complex numbers C. Let A 2 be a two dimensional affine space over C. The polynomials g in the ring Cy can be viewed as complex valued functions on A 2 by evaluating g at the points in A 2 .

* Let k be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k over k.

Let V be an algebraic set defined as the set of the common zeros of an ideal I in a polynomial ring over a field K, and let A = R / I be the algebra of the polynomials over V. Then the dimension of V is:

Let R be a ring, and let M be a left R-module.

On November 21, 2009, Flair returned to the ring as a heel on the " Hulkamania: Let the Battle Begin " tour of Australia, losing to Hulk Hogan in the main event of the first show by brassknuckles.

Let G be a group, written multiplicatively, and let R be a ring.

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