[permalink] [id link]

Every ring which is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.

from
Wikipedia

## Some Related Sentences

Every and ring

__Every__Boolean algebra ( A

**,**∧, ∨) gives rise to

**a**

__ring__( A

**,**+, ·) by defining

**a**+ b := (

**a**∧ ¬ b ) ∨ ( b ∧ ¬

**a**) = (

**a**∨ b ) ∧ ¬(

**a**∧ b ) ( this operation

**is**called symmetric difference in the case of sets and XOR in the case of logic ) and

**a**· b :=

**a**∧ b

**.**The

**zero**element of this

__ring__coincides

**with**the 0 of the Boolean algebra ; the multiplicative identity element of the

__ring__

**is**the 1 of the Boolean algebra

**.**

In Norse mythology

**,**Draupnir ( Old Norse " the dripper ")**is****a**gold__ring__possessed by the god Odin**with**the ability to multiply**itself**:__Every__ninth night eight new rings ' drip ' from Draupnir**,**each one of the same size and weight**as**the original**.**__Every__

**module**

**over**

**a**division

__ring__

**has**

**a**basis ; linear maps between finite-dimensional modules

**over**

**a**division

__ring__can be described by matrices

**,**and the Gaussian elimination algorithm remains applicable

**.**

__Every__objective

**has**

**a**different size

__ring__

**,**so for

**every**objective another condenser setting

**has**to be chosen

**.**

__Every__match must be assigned

**a**rule keeper known

**as**

**a**referee

**,**who

**is**the final arbitrator ( In multi-man lucha libre matches

**,**two referees are used

**,**one inside the

__ring__and one outside ).

* In any

__ring__R**,****a**maximal ideal**is**an ideal M that**is**maximal in the set of all proper ideals of R**,**i**.**e**.**M**is**contained in exactly 2 ideals of R**,**namely M**itself**and the entire__ring__R**.**__Every__maximal ideal**is**in fact prime**.**__Every__prime ideal P in

**a**Boolean

__ring__R

**is**maximal: the quotient

__ring__R / P

**is**an integral domain and also

**a**Boolean

__ring__

**,**so it

**is**isomorphic to the field F < sub > 2 </ sub >,

**which**shows the maximality of P

**.**Since maximal ideals are always prime

**,**prime ideals and maximal ideals coincide in Boolean rings

**.**

__Every__topological

__ring__

**is**

**a**topological group (

**with**respect to addition ) and hence

**a**uniform space in

**a**natural manner

**.**

__Every__polynomial in can be factorized into polynomials that are irreducible

**over**F

**.**This factorization

**is**unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the

__ring__of polynomials

**over**

**a**field

**is**

**a**unique factorization domain whose units are the nonzero constant polynomials ).

Every and which

__Every__new scandal

__which__would provide more `` copy '' for Marshall's pen would thus mean more publicity for Welch

**.**

__Every__dream

**,**and this

**is**true of

**a**mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought

**,**

**is**

**a**psychic phenomenon for

__which__no explanation

**is**available

**.**

__Every__family of Riviera Presbyterian Church

**has**been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in

__which__all members of the family participate

**.**

** Zorn's lemma:

__Every__non-empty partially ordered set in__which__**every**chain ( i**.**e**.**totally ordered subset )**has**an upper bound contains at least one maximal element**.**
*

__Every__continuous functor on**a**small-complete category__which__satisfies the appropriate solution set condition**has****a**left-adjoint ( the Freyd adjoint functor theorem ).__Every__telephone company

**,**whether large or small

**,**determines its own ANAC for each individual central office

**,**

__which__tends to perpetuate the current situation of

**a**mess of overlapping and / or spotty areas of coverage

**.**

__Every__field

**has**an algebraic extension

__which__

**is**algebraically closed ( called its algebraic closure ),

**but**proving this in general requires some form of the axiom of choice

**.**

__Every__year since 1972 the BVI

**has**hosted the Spring Regatta

**,**

__which__

**is**

**a**seven-day collection of sailing races throughout the islands

**.**

*

__Every__real Banach algebra__which__**is****a**division algebra**is**isomorphic to the reals**,**the complexes**,**or the quaternions**.**
*

__Every__unital real Banach algebra**with**no**zero**divisors**,**and in__which__**every**principal ideal**is**closed**,****is**isomorphic to the reals**,**the complexes**,**or the quaternions**.**__Every__character

**is**automatically continuous from A to C

**,**since the kernel of

**a**character

**is**

**a**maximal ideal

**,**

__which__

**is**closed

**.**

__Every__computer contains an internal clock that regulates the rate at

__which__instructions are executed and synchronizes all the various computer components

**.**

* Duality:

__Every__statement**,**theorem**,**or definition in category theory**has****a**dual__which__**is**essentially obtained by " reversing all the arrows ".__Every__context-sensitive grammar

__which__does

**not**generate the empty string can be transformed into an equivalent one in Kuroda normal form

**.**

According to

__Every__**,**one example may be " the myth of St**.**George " and other stories about saints battling dragons**,**__which__were " modelled no doubt in many cases on older representations of the creator and preserver of the world in combat**with**chaos ".__Every__argues that " the disparagement of myth in our own civilization " stems partly from objections to perceived idolatry

**,**objections

__which__intensified in the Reformation

**,**both among Protestants and among Catholics reacting against the classical mythology revived during the Renaissance

**.**

__Every__grammar in Chomsky normal form

**is**context-free

**,**and conversely

**,**

**every**context-free grammar can be transformed into an equivalent one

__which__

**is**in Chomsky normal form

**.**

__Every__four years

**,**during

__which__an extra 24 hours have accumulated

**,**one extra day

**is**added to keep the count coordinated

**with**the sun's apparent position

**.**

__Every__sequence can

**,**thus

**,**be read in three reading frames

**,**each of

__which__will produce

**a**different amino acid sequence ( in the given example

**,**Gly-Lys-Pro

**,**Gly-Asn

**,**or Glu-Thr

**,**respectively ).

__Every__time the boy had an injury

__which__caused him internal or external bleeding

**,**the Tsarina called on Rasputin

**,**and the Tsarevich subsequently got better

**.**

__Every__week before June 19

**,**the strip focuses on Garfield's birthday

**,**

__which__he dreads because of his fear of getting older

**.**

:::

__Every__Bill__which__shall have passed the House of Representatives and the Senate**,**shall**,**before it become**a**Law**,**be presented to the President of the United States ; If he approves he shall sign it**,****but**if**not**he shall return it**,****with**his Objections to that House in__which__it shall have originated ...0.111 seconds.