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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:

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## Some Related Sentences

Given and ring

*

__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**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**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__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 and R

__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__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 and proper

* Named entity recognition

**(**NER ):__Given__**a**stream**of**text, determine which items in**the**text map to__proper__names, such as people or places,**and**what**the**type**of**each such name**is****(**e. g. person, location, organization ).__Given__

__proper__combustion chamber design, pre-ignition can generally be eliminated by

__proper__spark plug selection,

__proper__fuel / air mixture adjustment,

**and**periodic cleaning

**of**

**the**combustion chambers.

__Given__

**the**abnormalities

**of**

**the**microvasculature

**and**other side effects

**of**diabetes, these wounds take

**a**long time to heal

**and**require

**a**specialized treatment approach for

__proper__healing.

Using tensor calculus,

__proper__time**is**more rigorously defined in general relativity as follows**:**__Given__**a**spacetime which**is****a**pseudo-Riemannian manifold mapped with**a**coordinate system**and**equipped with**a**corresponding metric tensor,**the**__proper__time experienced in moving between two events along**a**timelike path P**is**given by**the**line integral__Given__

**the**

__proper__

**conditions**, theory Y managers believe

**that**employees will learn to seek out

**and**accept responsibility

**and**to exercise self-control

**and**self-direction in accomplishing objectives to which they are committed.

__Given__

**that**

**the**chelandia appear originally to have been oared horse-transports, this would imply differences in construction between

**the**chelandion

**and**

**the**dromōn

__proper__, terms which otherwise are often used indiscriminately in literary sources.

__Given__

**a**

__proper__language, programs, including operating system servers, could import multiple interfaces

**and**combine them as

**if**they were objects native to

**that**language — notably C ++.

__Given__

**the**delicate nature

**of**their missions, RENEA reverted to "

__proper__" sniper rifles

**and**currently employs SAKO TRG-22

**and**TRG-42 rifles.

__Given__

**a**complete Boolean algebra B there

**is**

**a**Boolean-valued model denoted by V < sup > B </ sup >, which

**is**

**the**Boolean-valued analogue

**of**

**the**von Neumann universe V.

**(**Strictly speaking, V < sup > B </ sup >

**is**

**a**

__proper__class, so we need to reinterpret what it means to be

**a**model appropriately.

Given and ideal

__Given__two fractional ideals

**I**

**and**J, one defines their product IJ as

**the**set

**of**all finite sums

**:**

**the**product IJ

**is**again

**a**fractional

__ideal__.

__Given__

**a**subset V

**of**A < sup > n </ sup >, we define

**I**

**(**V ) to be

**the**

__ideal__

**of**all functions vanishing on V

**:**

__Given__

**a**homogeneous prime

__ideal__P

**of**, let X be

**a**subset

**of**P < sup > n </ sup >( k ) consisting

**of**all roots

**of**polynomials in P .< ref > The definition makes sense since

**if**

**and**only

**if**for

**any**nonzero λ in k .</ ref > Here we show X admits

**a**structure

**of**variety by showing locally it

**is**an affine variety.

__Given__

**that**we are only interested in what happens on shell, we would often take

**the**quotient by

**the**

__ideal__generated by

**the**Euler-Lagrange equations, or in other words, consider

**the**equivalence class

**of**functionals / flows which agree on shell.

__Given__

__ideal__

**conditions**, females can have up to five litters per year although reproduction becomes depressed in summer

**and**ceases altogether in times

**of**drought.

__Given__his spirituality, Boulogne-sur-Mer may have been an

__ideal__choice for

**a**home

**:**in addition to its fine churches,

**the**city also contained numerous religious schools

**and**charitable organizations.

"

__Given__**the**gift**of**this " divinization " in grace, "**a**new principle**of**energy ,"**and**with**the**support**of**" Christ's family ,"**the**Church, Escrivá states**that****the**difficult__ideal__**of**becoming**a**saint, another Christ,**is**" also easy.0.426 seconds.