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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.

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## Some Related Sentences

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__the state

**of**the stream leaving stage R

__be__denoted by

**a**vector Af and the operating variables

**of**stage R by Af

**.**

__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 unital

__Let__

**A**

**be**

**a**complex

__unital__

**Banach**

**algebra**in which

**every**non-zero

**element**x

**is**invertible (

**a**division

**algebra**).

__Let__

**C**

**be**the category

**of**vector spaces K-Vect

**over**

**a**field K and let D

**be**the category

**of**algebras K-Alg

**over**K ( assumed

**to**

**be**

__unital__and associative ).

__Let__

**be**the Cartan matrix

**of**the Kac-Moody

**algebra**

**,**and let q

**be**

**a**nonzero complex number distinct from 1

**,**

**then**the quantum group

**,**U < sub > q </ sub >( G ), where G

**is**the Lie

**algebra**whose Cartan matrix

**is**

**A**

**,**

**is**defined as the

__unital__associative

**algebra**

**with**generators ( where λ

**is**an

**element**

**of**the weight lattice

**,**i

**.**e

**.**for all i ), and and ( for simple roots

**,**), subject

**to**the following relations:

__Let__X

**be**any Lie

**algebra**

**over**K

**.**Given

**a**

__unital__associative K-algebra U and

**a**Lie

**algebra**homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U

**is**the universal enveloping

**algebra**

**of**X if it satisfies the following universal property: for any

__unital__associative K-algebra

**A**and Lie

**algebra**homomorphism f: X →

**A**< sub > L </ sub > there exists

**a**unique

__unital__

**algebra**homomorphism g: U →

**A**such that: f (-) = g < sub > L </ sub > ( h (-)).

__Let__

**A**and B

**be**two

**commutative**rings

**with**unity

**,**and let f:

**A**→ B

**be**

**a**(

__unital__)

**ring**homomorphism

**.**

Let and commutative

__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**.**0.275 seconds.