VNU JOURNAL OF SCIENCE. Mathematics - Physics t.XVIII n°l - 2002

ON TH E LIN EA R RADICAL OF LATTICES
N guyen D ue D at
F n cu liy oí' M ỉìLlicm nìics, College o f N n tui’fil sricnces. V N Ư IỈ
1. I n t r o d u c t io n
T lir radical theory is an im portant tool for stud ying th e structure and th e cliissiliration of algebraic structures. It attra cts large interest of many authors 1. 2. iỉ. r>.
7 rii
Proof. (I ) We have r ( L ) = n {/),ị/ f I } . where /),./ 6 / . are linear congruencies on
/.As well-known. L / r ị L ) i.s tlu* Mil>- , / / e iyj,T. i.e.. 1-J-j.T < WinS nffirinrcy. T ile proof is trivial.
Now. considering all th e linear congruence ơ jf> . i £ / , on L//>, wo liavc r { L €‘rỊĩ) —

\ n , (> i
/} \ n , ! Ç / } / p . M ay be {ơ ,l/ € Í} (iocs not contain all the linear
h tu those of
radicals o f rings.
3. Application
111 th is section wo present an application o f the linear radical to classifying modular

lattices. Here, for lattices, it is particular that p is r-sem i sim ple class. Therefore, the
classification problem is o f interest itself.
For exam ple, consider a m odular lattice A /. A s in E xam ple 2.5. we see that M ạ
is r-iadieal lattice.

On the other hand, the sublattices of M which are isomorphic to

A/.J prevent M from being d istribu tive. Since M / r ( M ) is distributive., each sublattice
isomorphic to M\\ belongs to on e of equivalence classes of r ( M ) .
Based upon t.ỈK' above reasons we arrive at th e stu d y of the particular modular
bit tiers form ulated in the following th eorem .
3 .1 .

T h e o r e m . Lot M !)(' a m o d u la r lã t t k r which is not distributive.

Let .4 ÌH' ỈÌ

sublattice o f M such tlmt:

Sim ilarly to case (I) we can deduce that there exist the maximal ideal ./ and the
maxim al filter F such th at I) € »/,.7 0 / 1 = 0 and .4 ç F, .7 f F =■ </>.
Now. for both cases (I) and (II) we shall prove that ./ u F —.Ì/
w v suppose that 3c* € A /, c Ệ J u F . First we prove the assertion:
(i) \ j £ J J V c £ F .
hulivd. if V / €
A

v \

V Í/} A ( / V •».)

(u < v)

= [O' A v ) V u] A ( / V u)
= [(/ A ?/) A ( / V u)] V u
= {[j A ( / V r ) | A [ / V ( j A r )|} V «

( a

< / V

(s o e (l).{ 2 ))

= { ( / V c) A [j A ! / V (j A c)]]} V V
= ( /V r ) A [(jA r )V (jV /)]V u
= [Ơ A c) v (j V / ) ] V a = u

It)

(./'Ac

o f r ( M ) containing /1 is 0XHCt.lv equal to .4. T he other classes are su b la ttice o f M . T hese
subiaftice* contain !)() sublnttice isom orphic to A/ 3 , therefore, th ey arc distributive.
T he proof o f theorem is com pleted. □
3 . 3 .R e m a r k . It is worth re m a rk in g that lattices, ill general, are not d istrib u tive l)OCciu>v
thi'Y may contain not only the su bla tticc M; ị . hut also Xr>- Therefore, the classification
ỊỉVoliìrm is difficult (sre E x a m p le 2.5).

References
1. S.A. Am itsur. A general theory o f radicals II. R adicals ill lin g s and bicategories,
2.
3.
4.
5.
G.
7.

A iiirr. J. Math., 7 6 (1 9 5 4 ). 100- 125.
T. Anderson. N. Divinsky. A. Sulinski. H ereditary radicals in assoc iative and alter­
native rings. Catiad. ./. Math.. 1 7 (1 9 6 5 ). 594 - 603.
V .A . A nclrunakevitdi. O n the Theory o f the lin e a r radicals. D A N U SSR . 1 1 3 (1 9 5 7 ).4 8 7 490.
G. Gratzcr. General lattice theory. A kadem ie -Verlag. B erlin, 1978.
A.G. Kurosh. Linear R adicals and Algebra* Math. Collection. 3 3 (7 5 )( 1953). 13-26.
A. G. Kurosh. Lectures on general Algebra. P h ys - M ath., P u b lish in g house. 1974.
F.A . Szsz. Radicals o f lin g s. Akadm iai Kiacl. B u d a p est, 1981.

TAP CHI KHOA HỌC DHQGHN. Toán - Lý. t.XVIII. n °l - 2002

VỀ CÁN TUYẾN TÍNH CỦA CÁC DÀN
Nguyền Đức Đạt
K h o a lo a n . D ụ i h ọ c K h o a học T ự nhiên - f ) I I Q ( i / l ủ N ộ i