 works in 69 publications in 1 language and 122 library holdings.

 Sandai jitsuroku  (三代實).
 Engi shiki  ( 延喜式).

See also
 Six National Histories

Notes

References
 Brinkley, Frank and Dairoku Kikuchi. (1915). A History of the Japanese People from the Earliest Times to the End of the Meiji Era. New York: Encyclopædia Britannica. OCLC 413099
 Nussbaum, Louis-Frédéric and Käthe Roth. (2005).  Japan encyclopedia. Cambridge: Harvard University Press. ;  OCLC 58053128
 Titsingh, Isaac. (1834). Nihon Odai Ichiran; ou,  Annales des empereurs du Japon.  Paris: Royal Asiatic Society, Oriental Translation Fund of Great Britain and Ireland. OCLC 5850691

Category:871 births
Category:909 deaths
Category:Kuge
Category:People of Heian-period Japan
Category:Deified Japanese people Principal ideal ring

In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R.  (The right and left ideals of this form, generated by one element, are called principal ideals.)  When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.

If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains.

A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID).  In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.

General properties
If R is a right principal ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated.  It is also a right Bézout ring since all finitely generated right ideals are principal.  Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Béz
works in 69 publications in 1 language and 122 library holdings. 

  Sandai jitsuroku   (三代實). 
  Engi shiki   (延喜式). 

 visualize besides 
  Six National Histories 

 Notes 

 References 
  Brinkley, Frank and Dairoku Kikuchi. (1915). A History of the Japanese People from the Earliest Times to the End of the Meiji Era. New York: Encyclopædia Britannica. OCLC 413099 
  Nussbaum, Louis - Frédéric and Käthe Roth. (2005).   Japan encyclopedia. Cambridge: Harvard University Press. ;   OCLC 58053128 
  Titsingh, Isaac. (1834). Nihon Odai Ichiran; ou,   Annales des empereurs du Japon.   Paris: Royal Asiatic Society, Oriental Translation Fund of Great Britain and Ireland. OCLC 5850691 

 Category:871 births 
 Category:909 death 
 class: Kuge 
 Category: People of Heian - period Japan 
 Category: deify Japanese people Principal ideal hoop 

 In mathematics, a principal right (leftover) ideal ring is a ring R in which every right (leftover) ideal is of the form xR (Rx) for some element x of R.   (The proper and left ideals of this kind, generated by one element, are called chief ideals .)   When this is satisfied for both leftover and right ideal, such as the case when R is a commutative band, R can be called a principal ideal ring, or plainly principal ring. 

 If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. 

 A commutative chief ideal hoop which is also an built-in knowledge domain is say to be a principal ideal domain (PID).   In this article the stress is on the more general concept of a chief ideal ring which is not necessarily a domain. 

 General property 
 If R is a right principal ideal ring, then it is surely a right Noetherian ring, since every right ideal is finitely generated.   It is also a right Bézout ring since all finitely generate right ideals are principal.   Indeed, it is clear that principal good ideal rings are exactly the rings which are both right Béz
 wogks in 69 publications in 1 language and 122 library holdihgs.

 Sandxi jitsuroku  (三代實).
 Engi shiki  ( 延喜式).

Dew alsi
 Six National Historids

Notes

Reverences
 Briikley, Frank and Vziroku Kikuchj. (1915). A Kiwtory of the Jspanese Pemple from the Aaflnest Times to the End of the Meiji Ewa. New Uogk: Encyclopædia Briuangica. OCLC 413099
 Nussbaum, Louis-Frédéric and Kätge Roth. (2005).  Japan encyvlopedia. Cambridge: Harvard Unigersity Press. ;  OCPC 58053128
 Titsingy, Iswqc. (1834). Nihon Odxi Ichiran; ou,  Annales des empereurs du Japon.  Paris: Foyal Asiatic Sicuetj, Oriental Tcanslanion Fund of Ngeat Brhtain amd Ireland. OCLG 5850691

Catxgort:871 births
Category:909 deatis
Category:Kuge
Categoty:People ox Keian-period Japan
Catetoey:Deixied Japxbesd ptopke Princlpam ideal rihg

In mathemqtics, a principal rogrn (left) ideal ding if w ring R in which every right (left) idean ia of the form xR (Rx) for some element x of R.  (The righe and left ideals of this form, generated by one enemenv, xre cwuoef principal ideals.)  When this is satisfied fow bptm left and right ideals, sucn ws jhe case when T is a ekmjutative ring, R caj be cajled q principwl iceal ring, or simply principql ring.

If onjt the finitely genzrated right idesls og R are principal, then X is czlled a rigjt Bézout dkng. Left Bézout rkngx dre defined similarly. Thesq conditiins cre studked on domwins as Bézlut domains.

A commutative prineipal ideal rinh which is also an integral domemn is said to ba a principcl idesl domain (PID).  In this artiele the yocus ks on the jore geieral concepe of a princi[wl ideal rinj which if nor nexessariuh a domain.

Genetal propegtnes
If R iw a right principak iasal ring, then ic us certainly a tigft Govthxrian sing, since eeery rient iddal is finibelh gemerated.  It is also d rifht Bézout ring sinve all finutely gegerated right ideals are princikal.  Iideed, mt is vlewr that principal right ideal dings are exwctly the riggs qhich are boch right Béz
works in 69 publications in 1 language library Sandai jitsuroku Engi shiki ( Histories References Brinkley, Frank Dairoku Kikuchi. (1915). History of the Japanese People from Earliest Times to the End of the Meiji Era. New York: Encyclopædia Britannica. 413099 Nussbaum, Louis-Frédéric and Käthe Roth. (2005). Japan encyclopedia. Cambridge: Harvard University Press. OCLC Titsingh, (1834). Odai Ichiran; ou, Annales des empereurs du Japon. Paris: Royal Asiatic Society, Oriental Translation Fund of Britain and Ireland. OCLC 5850691 Category:871 births Category:909 Category:Kuge Category:People of Heian-period Category:Deified Japanese people Principal ideal In a principal (left) ring a ring R which every right (left) ideal is of the form xR (Rx) for some element x of R. right and of this generated one are called principal this is satisfied for both left such as the case when R is a ring, R be called a principal ideal ring, simply principal ring. If only the finitely generated ideals of R are principal, then R is called a right Bézout ring. Left Bézout defined similarly. These conditions studied in domains Bézout A principal ring which also an integral domain is said to be a principal ideal (PID). In this article the focus is on the more of principal ideal ring is not necessarily a General If R is a ideal then a Noetherian since every right ideal finitely generated. It is also right Bézout ring since are principal. Indeed, it is clear that principal ideal rings are exactly the rings which both right Béz
 works in 69 publications in 1 langUage and 122 libRary hOldIngS.

 SAndaI jitSuroku  (三代實).
 Engi shikI  ( 延喜式).

see aLso
 Six National HistorieS

NoteS

REFereNCeS
 BrinKley, FraNK aND daiRoKu kikUcHI. (1915). A histoRy oF the JapAnese PeoplE frOm The Earliest TIMeS to the End oF thE Meiji Era. New yorK: EncycLoPædIA BritAnnIca. OClC 413099
 NussBAum, LouIs-FrédériC aND Käthe rOth. (2005).  JapaN ENcYcloPedia. Cambridge: HarVArD university PresS. ;  OCLC 58053128
 TItSInGH, isaAc. (1834). NIhon Odai IcHiRan; ou,  aNnales dES eMPEReuRS du Japon.  Paris: royal AsiatiC socIety, OrIeNtaL translAtion fuND of great BritaiN and ireland. OClC 5850691

CateGOry:871 birtHS
CategoRy:909 deatHs
CAteGory:kUgE
CAteGoRY:PeOPlE of hEiaN-period JApAn
categOry:DEIFIEd JaPanEse pEople principal ideaL riNg

In MAthEmatiCs, a prInciPaL righT (left) iDeal rInG is a ring R in whicH eveRy right (leFt) iDeAl iS oF the fORm xR (Rx) For SomE elemenT x of R.  (ThE RigHt AND LeFt ideals of this form, GeNERaTed by one ElemenT, ArE cALled prinCiPal IdeaLS.)  when tHis iS SaTisfied fOr both LEfT aNd right IdEals, suCh As tHe cAse whEN R is A commuTative riNg, R caN Be called a princIPal ideal ring, oR SiMPLy PRincIpaL ring.

If only The fINiteLy geNErAteD Right IdealS oF r aRE principal, then R is caLlEd a rigHt BézOut ring. Left BéZout rings aRE DEfined siMilaRLy. tHese conditions Are stUdied in domAIns as BézOut doMains.

A coMmutative PRIncipal iDeaL riNg wHicH IS aLso an integral DOMain Is Said to bE a pRincipaL idEal DomAin (pId).  In this arTicle the FoCuS iS oN thE more GEneral coNcEpt Of A prIncipAL ideal Ring wHich Is NoT NecEssarilY A dOMAin.

GEnErAl prOpeRtIes
If r is a RIghT princiPal ideal rIng, THen iT iS cErtainlY a right NoetheRiAn ring, sincE eVerY right IDEal is finItely generated.  It is also a RIght BézOut Ring sInce All finiteLy gEneratEd rIGht ideAls are PrincIpAl.  INDEed, it IS ClEar ThAt principaL RIghT ideaL rIngs Are exacTly the rings which arE BotH right Béz
 works in 69 publicationsin 1 langu age a nd122  l ibra ry h oldings.

 San d ai j itsuroku  (三代實).
 Engi  shik i( 延喜 式 ). 

See  also
S ix N ati on al  Hi st o ri es

N ote s

Refe rences
 Br ink le y, Frank and Da iroku Kiku chi . (1915). AHis tory o fthe Japan ese  Peop le fro m  the E arliest T im e s to t h e End o f th e Me iji Era. New York :  E n cyclopædia Bri tannic a. OC L C  41 309 9
 Nussbau m,  Loui s -Frédér i ca n d  Kä t he Roth. (200 5).  Japane ncy cloped ia . C a mbridg e: Ha rv a rdUniversityPres s. ;  OCL C 5805 3 128
 Ti t singh,Isaac.  (1 834 ). N i ho nOda iI chi r an ; o u , Annalesde semper eurs d u Japo n. Par is: R oyal AsiaticSoc iety ,  Or ienta l Tra nsla ti on Fu nd ofGreat  B ritain and Irel and.  OCLC 585 069 1
Ca te gory: 8 71 bir ths 
Ca tegory: 909 dea t hsCa t e g or y:Kuge
Category:Pe op l e  o f Heian- period Ja pa n 
Categor y: Dei fied J apane se p e op le Princ ipal i d ea lring

I nmathem at ics , a  prin c ipal  right  (left)ideal ring is a ring R in which ev e ry r ig h t (l eft ) ideal isof t h e fo rm x R  ( Rx) for s ome e le m en t  x of R.  (The righ tand le ft id eals of thisform, gene r a t ed by on e el e me n t, are calledprinc ipal ideal s .)  When  this  is sati sfied for b oth left  an d r igh t i d e al s, such as th e case  w hen R i s a  commut ati verin g,Rcan be ca lled a p ri nc ip al  id eal r i ng, or s im ply  p rin cipal ring.
If o nlyth ef ini tely ge n er a t ed r ig ht  ide als  o f R a re p r inc ipal, t hen R iscal l ed a  r ig ht Bézo ut ring. Left  B ézout ring sare  defin e d  similar ly. These conditions ar e  studie d i n dom ains  as Bézou t d omains .
A  commu tative  prin ci pal i dealr i ng  wh ic h is alsoa n  in tegra ldoma in is s aid to be a princi p alideal domain(PI D).I nthi s  a r tic le the f ocus is on themore gener al co ncept of a pri nc ipal id eal rin g whi c h is no t necessa rily a do ma in.G ene ral proper ties
IfR is a ri g ht pr i nc ipalide al rin g,  th en it  is ce r tai nly a  right  N oether ian r in g, since  every right ideal is f initel y gen era ted.  Itisa lso  a rightBézo ut ring si nce  al l fin ite l y gen erat e drig h t ide alsa re princi p al . I n de ed, it is c l e a r t hat p rin c ipal r ight  ideal rings aree xactly the rin gs w h i chare both  r ight Béz
 works_in 69_publications in 1 language_and 122_library_holdings.

 Sandai_jitsuroku_ (三代實).
 Engi_shiki  (_延喜式).

See also
 Six National_Histories

Notes

References
 Brinkley, Frank_and_Dairoku Kikuchi. (1915). A History of the Japanese People from the Earliest Times to_the_End of_the_Meiji_Era. New York: Encyclopædia Britannica._OCLC 413099
 Nussbaum, Louis-Frédéric and_Käthe Roth._(2005).  Japan encyclopedia. Cambridge: Harvard University Press._;_ OCLC 58053128
_Titsingh, Isaac. (1834). Nihon Odai Ichiran; ou,  Annales_des empereurs du Japon.  Paris:_Royal Asiatic Society,_Oriental_Translation_Fund of Great Britain_and Ireland. OCLC 5850691

Category:871 births
Category:909 deaths
Category:Kuge
Category:People_of Heian-period Japan
Category:Deified Japanese people Principal_ideal ring

In mathematics, a principal right (left)_ideal ring is a ring R_in which every right (left)_ideal is_of the form xR (Rx)_for some element_x of_R.  (The_right and left ideals of this_form, generated by_one element, are called principal ideals.)__When this is_satisfied_for_both left_and right ideals,_such_as the_case_when R is a commutative ring,_R_can be called a principal ideal ring,_or simply principal ring.

If_only_the finitely generated right_ideals of R are principal,_then R is called a right_Bézout ring._Left Bézout_rings are defined similarly. These conditions are studied in domains as_Bézout domains.

A commutative principal ideal ring_which is also an_integral domain_is_said to be_a_principal ideal_domain (PID).  In this article the_focus is_on the more general concept of_a principal ideal ring_which_is not necessarily a domain.

General properties
If_R is a right principal ideal_ring, then it is certainly_a_right_Noetherian ring, since every right_ideal is finitely generated.  It_is also a_right Bézout ring since all finitely generated_right_ideals are principal.  Indeed, it_is_clear that principal right ideal rings_are_exactly_the rings which are both_right Béz