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Page "Algebra (ring theory)" ¶ 2
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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 be a ring and G be a monoid.
* 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 < sub > x </ sub >, defined by
Let R < sup > 2n </ sup > have the basis
If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 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 < sup > m </ sup > → R < sup > n </ sup > a short C < sup >∞</ sup >- embedding ( or immersion ) into Euclidean space R < sup > n </ sup >, where n ≥ m + 1.
Let U and V be two open sets in R < sup > n </ sup >.

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 commutative
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 g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements ( sometimes called Cartan subalgebra ) and let V be a finite dimensional representation of g. If g is semisimple, then g = g and so all weights on g are trivial.
Let A be a commutative Banach algebra, defined over the field C of complex numbers.
Let A be a commutative ring.
Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free-module.
Let R be any ring, not necessarily commutative.
Let R be a commutative ring and let A and B be R-algebras.
Let be a commutative ring with 1, e. g. ( Instead we can take to be a field and can replace by the field ).
Let K be a fixed commutative ring.
Let A be a superalgebra over a commutative ring K. The submodule A < sub > 0 </ sub >, consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a subalgebra of A, naturally called the even subalgebra.
Let R be a commutative superring.
Let R be a ( Noetherian, commutative ) regular local ring and P and Q be prime ideals of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra.
Let A be a commutative Noetherian ring with unity.
Let R and S be commutative rings and φ: R → S a ring homomorphism.
Let R be a commutative ring with prime characteristic p ( an integral domain of positive characteristic always has prime characteristic, for example ).
Let R be a commutative ring with unity, and let M, N and L be three R-modules.
Let M be a commutative monoid.
Let R be a commutative ring with identity 1.
* Let R ⊂ S be an integral extension of commutative rings, and P a prime ideal of R. Then there is a prime ideal Q in S such that Q ∩ R = P. Moreover, Q can be chosen to contain any prime Q < sub > 1 </ sub > of S such that Q < sub > 1 </ sub > ∩ R ⊂ P.
Let A ⊆ B be an extension of commutative rings.
Let A and B be two commutative rings with unity, and let f: A → B be a ( unital ) ring homomorphism.

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