数理解析研究所講究録
第 1915 巻 2014 年 69-73

69

The Structure of Fully Prime Rings Under Ascending Chain Conditions
Hisaya Tsutsui
Department of Mathematics
Embry-Riddle Aeronautical University
Prescott, AZ USA
E-mail address: Hisaya.Tsutsui@erau.edu

O. Introduction

Definition A ring$R$(possibly without identity) in which every ideal is prime $(i.e$ . every

proper ideal is a prime ideal and (hence)$R$

is a prime ring) is called a fully prime ring.

The purpose ofmy talk at the conference was to introduce the study of fully prime rings andpresent a few open problems to a wider range of audience. In this paper, a brief survey ofresults on the structure of fully prime rings with several examples is presented.
Our investigation for the structure of fully prime rings was initially motivated by a well-known fact that a commutative fully prime ring with identity is a field. We first published aseries ofpapers on the subject in 1994 and 1996 (Blair-Tsutsui[l], Tsutsui [7]). Conditions
similar to the fully prime condition have received attention in literature. Hirano [3] studied
those rings in which every ideal is completely prime. Courter [2] studied those rings in which

every ideal is semiprime, and Koh [6] studied those rings in which every right ideal is prime.
More recently, Hirano-Tsutsui [5] studied rings in which every ideal is -primary, and$n$

Hirano-Poon-Tsutsui [4] studied those rings in which every ideal is weakly prime. For many
years after our initial publications, our focus has been to determine the structure ofnoncommutative right and left Noethrian fully prime ring.
Throughout this article, we assume a ring to be associative but not necessarily commutative.
Due to the consideration that an ideal ofa ring being a ring of its own, we do not assume theexistence of a multiplicative identity on a ring unless otherwise so stated.
1. A few basic theorems on fully prime rings

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Theorem 1 (Blair-Tsutsui [1]). The center ofa hlly prime ring is either a field or zero, and aring$R$

is fully prime if and only ifevery (two sided) ideal $ofR$ is idempotent and the set ofideals $ofR$ is totally ordered under inclusion.
In particular, as is well known, a commutative $m[ly$ prime ring with identity is a field.

Examples ofa non-commutative fully prime ring include the ring ofendomorphisms$Hom_{D}(V,V)$ofa vector space$V$over a division ring D. (This can easily be verified by thetheorem above.) Denote the cardinality of a denumerable set by
let $\dim_{D}V=\aleph_{n-1}$ . Then is $fu$ ]$ly$prime with exactly$I_{N_{0}}=\{f\in Hom_{D}(V,V)|\dim f(V)<\aleph_{n}\},$

first limit ordinal, then $Hom_{D}(V,V)$ is a$k$, and for any integer$n\geq 1,$

non-zero proper ideals$n=0,2,\ldots,k-1$$fi\iota 11y$$N_{0}$. If$\dim_{D}V=\aleph_{a}b$where$\omega_{0}$is theprime ring that has countably many ideals.

Every right ideal, and hence every ideal of a regular ring is idempotent and for a regular self-injective rings , an ideal$P$

is prime if and only if RIP is totally ordered. Since there exists aregular self-injective ring$T$with a prime ideal

$R$

$P$

such that the set of ideals of $TlP$ is not well-ordered, the set of ideals of a $fiJ[1y$ prime ring is not necessarily well-ordered.
Blair-Tsutsui[l] gives an example of a fully prime ring (with exactly one nonzero properideal) which is not primitive. A ring

every proper ideal of
$I$

$R$

$R$

is called fully isomorphic if $RlI$ is isomorphic to$R$for. Non-commutative hlly isomorphic rings are fully prime, and there

exists an example of a non-commutative fully isomorphic ring with infinitely many ideals,
from which an example ofa non-primitive fully prime ring with identity with infinitely many
ideals may also be constructed.
Let$F$

be a field, and$[^{A}0bcbc0$

where

$A$

$R$

be the set ofall infinite matrices over$F$

that have the form$]$

is an arbitrary $2n$ by $2n$ matrix and and are any elements of . Then
$b$

$c$

$F$

$R$

is a prime

ring all ofwhose ideals are idempotent that contains non-prime ideals.
Problem 1. Under what conditions, would a prime ring in which every ideal is idempotent bea fully prime ring?

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Theorem 2 (Blair-Tsutsui [1]). Let$R$

be a fully prime ring. Then every ideal $ofR$ is fully

prime when it is considered as a ring without identity. Every ideal ofan ideal of is an ideal
$R$

$ofR$

. Further, a proper ideal $ofR$ cannot be a ring with identity.

Ifis a proper ideal ofa fully prime ring$R$

with identity and$F$

is a subfield ofthe center of, then $S_{P}=F+P$ is a fully prime ring whose maximal ideal$P$

is also a maximal right and

$P$

$R$

left ideal. Further, proper ideals $ofS$ are precisely those ideals $ofR$ that are contained in $P.$
Theorem 3 (Blair-Tsutsui [1]). $S_{P}=F+P$ is right primitive ifand only $ifR$ is right primitive.
$S_{P}$is semiprimitive if and only if is semiprimitive.
$R$

2. Right Noetherian Fully Prime Rings
In this section, we assume that a ring has a multiplicative identity.
Theorem 4 (Blair-Tsutsui [1]). A fully prime, right fully bounded right Noetherian ring issimple Artinian.
More generally, a prime right fully bounded right Noetherian ring, all ofwhose ideals areidempotent, is a simple Artinian ring (Tsutsui [7]).
As shown in Blair-Tsutsui [1], a fully prime ring is, in general, not semiprimitive. By

Nakayama’s lemma, it is evident, however, that a right Noetherian fully prime ring issemiprimitive. Further, an induction argument on the Krull dimension yields the following
result.
Theorem 5 (Tsutsui [8]). A fully prime ring with right Krull dimension is semiprimitive.
Let $A_{1}(k)be$ the first Weyl algebra over a field ofcharacteristic O. Then $S=k+xA_{1}(k)is$ aright and left Notherian fully prime ring with exactly one nonzero proper ideal
Noetherian$fi_{J}11y$$xA_{1}(k)$. Thus,

prime ring is not necessarily a simple ring.

Theorem 6 (Tsutsui [8]). Let $S=k+xA_{1}(k)$ where the characteristic ofidealizer of any maximal right ideal $M$ of$S$

diﬀerent from$xA_{1}(k)$$k$

is zero. Then theis not a fully prime ring.

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Whether there exists a fully prime right Noetherian ring with more than one nonzero properideals remains a tantalizing open question. Let $R=k+xA_{1}(k)\otimes_{k}(k+xA_{1}(k))$ . Then
exactly two nonzero proper ideals$xA_{1}(k)\otimes_{k}xA_{1}(k)$and $x4(k)\otimes_{k}(k+xA_{1}(k))$ and

$R$

$R$

has

is afUlly prime ring.
Problem 2. Is $R=k+xA_{1}(k)\otimes_{k}(k+xA_{1}(k))$ a right Noetherian ring?
Theorem 7. (Tsutsui [8]). A fully prime ring with identity that has a right Krull dimension

and a minimal nonzero ideal is a right primitive ring.
Let

$R$

be a ring with identity and$\mathfrak{M}_{R}$be the class of all right -modules. The class of$R$modules that are closed under taking factors, sum, extensions, and submodules is called ahereditary torsion class. A hereditary torsion class is called a TTF-class if it is also closed
under taking direct products. It is well-known that$s^{\sim}=\{M\in \mathfrak{M}_{R}|MI=0\}$prime ring

$R$

$I$

$R$

$P$

ofa fUllyis idempotent and the set of ideals is linearly ordered, the TTF-classes
ofa ﬃlly prime ringis a fully prime ring. It is evident thatsmallest TTF-class Containing$\overline{\mathfrak{A}}=K_{\bigcap_{0\neq P\triangleleft R}P}$

is a TTF-class ifand only if

for some idempotent ideal of . Since every ideal$K_{P}=\{M\in \mathfrak{M}_{R}|MP=0\}$$R$$s^{arrow}$$\mathfrak{A}$$\mathfrak{A}$$R$

are linearly ordered. Let$\mathfrak{A}=\bigcup_{0\neq P\triangleleft R}K_{P}$is a hereditary torsion class. Let. Then since$K_{P}\subseteq\overline{\mathfrak{A}}$. Considering a module representation of$\overline{\mathfrak{A}}$wherebe thefor every nonzero ideal $P,$$R$

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

we are strongly hopeful toprove, by contradiction, that a fully prime ring is primitive ifand only if it is semiprimitive.
This result, ifaﬃrmative, will in particular imply that a fully prime ring is primitive if it isvon Neumann regular. It would also imply that a fully prime right Noetherian ring is primitive
with finitely many ideals.
Acknowledgement
I sincerely thank Professor Tsunekazu Nishinaka for the invitation and his superb hospitality

during the conference and our research meetings in Japan this year.

References[1] W.D. Blair and H. Tsutsui, Fully Prime Rings, Comm. Algebra (1994).
[2] R. Courter, Rings all ofwhosefactor rings are semiprime. Canad. Math Bull (1969).

73[3] Y. Hirano, On rings all ofwhose factor rings are domain. J. Austral. Math.Soc. (1993).
[4] Y. Hirano, E. Poon, and H.Tsutsui, On rings in which every ideal is weakly prime,Bulletin, KMS Vo147, No.5 (2010).
[5] Y. Hirano and H.Tsutsui, Fully -primary rings, Comm. Algebra (2009).
$k$[6] K. Koh, On one-sided ideals ofprime type, Proc. Amer. Math Soc. (1971).
[7] H. Tsutsui, Fully Prime Rings, Comm. Algebra 22 (1996).

$\Pi$[8] H. Tsutsui, On rings in which every ideal is prime, Contemporary Ring Theory (2011).