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Let R be a commutative ring.

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## Some Related Sentences

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__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**.**
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__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

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__( 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 and be

__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__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__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__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__it

__be__granted then that the theological differences in this area between Protestants and Roman Catholics appear to

__be__irreconcilable

**.**

__Let__us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may

__be__''

**.**

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__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__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__

**be**

**a**

__commutative__

**ring**with 1, e

**.**g

**.**( Instead we can take to

**be**

**a**field and can replace by the field ).

__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**( 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__

**R**

**be**

**a**

__commutative__

**ring**with prime characteristic p ( an integral domain of positive characteristic always has prime characteristic, for example ).

*

__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 and B

**be**two

__commutative__rings with unity, and let f: A → B

**be**

**a**( unital )

**ring**homomorphism

**.**

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