The Structure of Fully Prime Rings Under Ascending Chain Conditions (Logics, Algebras and Languages in Computer Science)

The

Structure of Fully Prime Rings Under Ascending Chain Conditions

HisayaTsutsui Departmentof Mathematics Embry-Riddle Aeronautical University

Prescott,AZ USA

E-mail address:[email protected]

O. Introduction

Definition A ring$R$ (possiblywithoutidentity)inwhichevery idealis prime _$(i.e$. every properideal isaprime ideal and(hence)$R$ isaprimering) iscalledafully prime ring.

The

purpose

ofmytalk atthe conference

was

tointroducethestudy offully prime rings and present

a

fewopenproblems to

a

wider

range

ofaudience. Inthispaper,

a

brief

survey

of resultsonthe structure offullyprime rings with severalexamples is presented.

Ourinvestigationfor the structure of fullyprime ringswas initiallymotivated bya

well-known fact thatacommutativefullyprime ringwith identityisa field. Wefirst publisheda

seriesofpapers

on

the subject in 1994 and 1996(Blair-Tsutsui[l], Tsutsui [7]).Conditions similartothe fullyprime condition have receivedattentionin literature. Hirano [3] studied those ringsinwhicheveryideal is completelyprime. Courter[2] studied those ringsin which

everyidealis semiprime, and Koh [6] studied thoserings in which everyrightideal is prime. More recently,Hirano-Tsutsui [5] studiedrings in which every ideal is $n$-primary, and

Hirano-Poon-Tsutsui [4] studied those ringsin whicheveryidealisweaklyprime. Formany

yearsafter

our

initialpublications,

our

focus has been todetermine the structureof noncommutativeright and left Noethrian fullyprime ring.

Throughout thisarticle,

we

assume

a

ringtobeassociativebutnotnecessarilycommutative. Due to the considerationthat

an

idealofaring being

a

ringofitsown,

we

do not

assume

the existenceofamultiplicative identityon aringunlessotherwise so stated.

Theorem 1 (Blair-Tsutsui [1]). The center ofa hllyprimeringiseither

a

field

or

zero,and

a

ring$R$ is fully primeifand onlyifevery(two sided)ideal $ofR$ is idempotentand thesetof

ideals$ofR$is totally_{orderedunder inclusion.}

Inparticular,

as

is wellknown,

a

commutative $m[ly$prime ring with identity is

a

field.

Examples ofanon-commutativefullyprimering include thering ofendomorphisms

$Hom_{D}(V,V)$ofa vector

space

$V$

over a

divisionringD.(This

can

easily be verifiedby the

theoremabove.)Denotethe cardinality of

a

denumerablesetby $N_{0}$, and foranyinteger $n\geq 1,$

let $\dim_{D}V=\aleph_{n-1}$

.

Then is $fu$]$ly$primewithexactly $k$

non-zero

properideals

$I_{N_{0}}=\{f\in Hom_{D}(V,V)|\dim f(V)<\aleph_{n}\},$ $n=0,2,\ldots,k-1$

.

If$\dim_{D}V=\aleph_{a}b$where $\omega_{0}$ is the

firstlimitordinal,then $Hom_{D}(V,V)$ is

a

$fi\iota 11y$prime ring that has countablymanyideals.

Everyrightideal, and henceeveryideal ofaregularring isidempotentandfor

a

regular self-injectiverings$R$,

an

ideal $P$is prime if andonly ifRIP is totally ordered. Since there exists

a

regularself-injectivering $T$with

a

primeideal$P$such that thesetof ideals of$TlP$isnot

well-ordered,the setof ideals of

a

$fiJ[1y$prime ring isnot necessarilywell-ordered.

Blair-Tsutsui[l] gives

an

example ofafully prime ring(withexactly

one

nonzero

proper

ideal) whichisnotprimitive. A ring$R$is called fully isomorphic if$RlI$isisomorphicto$R$for

everyproperideal$I$of$R$

.

Non-commutative hlly isomorphicrings

are

fullyprime,and there

exists

an

exampleof

a

non-commutativefully isomorphic ring with infinitelymanyideals, from which

an

example ofa non-primitive fully prime ringwith identitywithinfinitely many idealsmayalsobeconstructed.

Let$F$be

a

field, and$R$be thesetofall infinitematrices

over

$F$that havetheform

$[^{A}0bcbc0$ $]$

where$A$ is

an

arbitrary$2n$by$2n$matrix and $b$and$c$

are

anyelements of$F$

.

Then$R$ is

a

prime ringallofwhose ideals

are

idempotent that contains non-primeideals.

Problem 1. Underwhatconditions,wouldaprime ring inwhichevery idealis idempotent be

Theorem2 (Blair-Tsutsui [1]). Let$R$be

a

fullyprime ring. Thenevery ideal_$ofR$is fully

prime when it is consideredas aring without identity. Every ideal ofan ideal of$R$is

an

ideal

$ofR$

.

_Further,

a

proper_ideal$ofR$_cannotbe

a

ring_{withidentity.}

If $P$is

a

proper

ideal ofa fullyprime ring$R$with identity and$F$is

a

subfield ofthecenterof

$R$,then

$S_{P}=F+P$ is

a

fullyprime ring whose maximal ideal$P$is also

a

maximalright and leftideal.Further, properideals$ofS$

are

_{preciselythose ideals}$ofR$_that

are

_containedin$P.$

Theorem3 (Blair-Tsutsui [1]). $S_{P}=F+P$ is right primitive ifand only$ifR$is right primitive.

$S_{P}$ is semiprimitiveif and only if$R$is semiprimitive.

2. Right Noetherian Fully Prime Rings

Inthis section,

we assume

that

a

ring has

a

multiplicative identity.

Theorem4(Blair-Tsutsui [1]). Afullyprime, right fully bounded right Noetherian ring is simpleArtinian.

More generally, aprime rightfully bounded right Noetherianring, all ofwhose idealsare

idempotent, is

a

simple Artinian ring(Tsutsui [7]).

Asshownin Blair-Tsutsui [1],

a

fullyprime ringis, in general, notsemiprimitive. By Nakayama’slemma,it isevident, however, thatarightNoetherian fullyprime ring is semiprimitive.Further,

an

inductionargument

on

the Krulldimensionyields the following result.

Theorem5(Tsutsui [8]). A fullyprime ringwith right Krull dimension is semiprimitive.

Let $A_{1}(k)be$thefirst Weyl algebraover

a

fieldofcharacteristic O. Then _{$S=k+xA_{1}(k)is$}a

right and leftNotherian fullyprime ringwith exactly

one nonzero

proper

ideal $xA_{1}(k)$

.

Thus,

Noetherian$fi_{J}11y$prime ring isnotnecessarilyasimple ring.

Theorem6(Tsutsui [8]). Let $S=k+xA_{1}(k)$where the characteristic of$k$ is

zero.

Thenthe idealizer ofanymaximal right ideal$M$of$S$different from _{$xA_{1}(k)$}isnot

a

fully primering.

Whether thereexists

a

fullyprime rightNoetherianringwith

more

than

one nonzero

proper

idealsremains

a

tantalizingopenquestion. Let $R=k+xA_{1}(k)\otimes_{k}(k+xA_{1}(k))$

.

Then $R$ has

exactly two

nonzero

properideals $xA_{1}(k)\otimes_{k}xA_{1}(k)$ and $x4(k)\otimes_{k}(k+xA_{1}(k))$and$R$is a fUlly prime ring.

Problem 2. Is $R=k+xA_{1}(k)\otimes_{k}(k+xA_{1}(k))$

a

right Noetherian ring?

Theorem 7.(Tsutsui [8]). Afully prime ring with identity that has

a

rightKrull dimension and

a

minimal

nonzero

ideal is arightprimitivering.

Let $R$ bearing with identity and $\mathfrak{M}_{R}$bethe class of allright$R$-modules. The class of

modulesthat

are

closed under takingfactors, sum, extensions, and submodules is called

a

hereditarytorsion class. Ahereditary torsion class is calledaTTF-classifit is also closed undertaking direct products. Itis well-known that $s^{arrow}$ is

a

TTF-class ifand onlyif

$s^{\sim}=\{M\in \mathfrak{M}_{R}|MI=0\}$for

some

idempotent ideal$I$of$R$

.

Since

every

ideal $P$ ofafUlly

prime ring $R$ is idempotent and theset ofidealsis linearlyordered,the TTF-classes

$K_{P}=\{M\in \mathfrak{M}_{R}|MP=0\}$ofa ffilly prime ring $R$

are

linearly ordered. Let

$\mathfrak{A}=\bigcup_{0\neq P\triangleleft R}K_{P}$where

$R$ is afullyprimering. Itisevident that $\mathfrak{A}$isahereditarytorsionclass. Let $\overline{\mathfrak{A}}$

bethe smallest TTF-class Containing$\mathfrak{A}$

.

Thensince $K_{P}\subseteq\overline{\mathfrak{A}}$

forevery

nonzero

ideal$P,$

$\overline{\mathfrak{A}}=K_{\bigcap_{0\neq P\triangleleft R}P}$

.

Considering

a

module representationof

$R$ in $\overline{\mathfrak{A}}_{\rangle}$

we are

stronglyhopefulto prove,bycontradiction, that

a

fullyprime ring is primitiveifand only ifitissemiprimitive. Thisresult, ifaffirmative, will in particular implythat

a

fullyprime ring is primitiveifit

is

von

Neumannregular. Itwould alsoimply that

a

fullyprime right Noetherian ring isprimitive withfinitelymanyideals.

Acknowledgement

IsincerelythankProfessor Tsunekazu Nishinakafor theinvitationandhis superb hospitality duringthe conferenceand

our

research meetings inJapanthisyear.

References

[1] W.D.BlairandH. Tsutsui, Fully Prime Rings, Comm. Algebra(1994).

[3] Y. Hirano, Onringsall

_ofwhose

_factor

ringsaredomain. J.Austral.Math.Soc. (1993).

[4] Y. Hirano, E.Poon, and H.Tsutsui, Onrings inwhicheveryidealisweaklyprime,

Bulletin, KMSVo147,No.5 (2010).

[5] Y. HiranoandH.Tsutsui,Fully$k$-primaryrings, Comm.Algebra(2009).

[6] K. Koh, Onone-sided idealsofprime type, Proc. Amer. Math Soc. (1971). [7] H. Tsutsui,Fully Prime Rings$\Pi$, Comm. Algebra22 (1996).