Union, Intersection, Product and Direct Product of Prime Ideals

Transcription

1 Globl Jourl of Pure d Appled Mthemtcs. ISSN Volume 11, Number 3 (2015), pp Reserch Id Publctos Uo, Itersecto, Product d Drect Product of Prme Idels Bdu.P (1), Dr.Srl.Y (2), Dr.Mdhusudh Ro.D (3), Dr.Ajeyulu.A (4) 1 Reserch Scholr, Deprtmet of Mthemtcs, K.L.Uversty, A.P, Id. 2 Fculty of Mthemtcs, K.L.Uversty, A.P, Id. 3 Fculty of Mthemtcs, VSR & NVR College, Tel, A.P, Id. 4 Retred HOD, Deprtmet of Mthemtcs, VSR & NVR College, Tel, A.P, Id. srlyell1970@gml.com Abstct The m gol of ths pper s to tte the oto of product d drect product of prme dels. We develop here some propertes of prme dels. It s show tht f product of fte set of prme rght dels of terry semgroup T s prme, the the product del cots t lest oe of the gve dels. It s lso show tht the tersecto of set of prme rght dels of terry semgroup T s prme rght del of T f d oly f t s prme rght del of the uo of the gve dels. Key words : Idel, Prme del d Drect product. Itroducto Prme dels of terry semgroups ply very mportt role terry semgroup. Severl uthors hve wored o ths mportt topc. These prme dels ws studed by Hle Brdely Grmble [3], Shbr [8] d Scwrtz [7]. The otos of terry semgroup s turl geerlzto of terry semgroup. The oto of del ply very mportt role to study the lgebrc structure. I Soso [9] d Kr [5] studed del theory terry semgroups. I Lehmer [6] developed the theory of terry semgroups. I ths pper we study some terestg propertes of prme dels of terry semgroups. Defto 2.1 A left del P of terry semgroup T s sd to be prme f, b, c T d bc P jotly mply tht ether Por b Por c P. Clerly, prme del mght be defed equvletly s del whose complmet s ether empty ( cse the del T tself, whch s obvously prme) or subterry semgroup of T.

2 1664 Bdu. P It s lso cler tht left zero elemet of T s prme rght del of T f d oly f t hs o proper dvsors. Theorem 2.2 The uo of rbtrry collecto of prme rght dels of terry semgroup T s prme rght del of T. Let R be collecto of prme rght dels of T, where rges over dex set M of rbtrry crdlty. The f, b, c T d bc P we must hve bc P for some M. But P s prme, whece ether P or b P or c P. Therefore ether P or b P or c P whece P s prme. Theorem 2.3 If product of fte set of prme rght dels of terry semgroup T s prme, the the product del cots t lest oe of the gve dels. Let R1, R2,... R be prme rght dels of T, d let product R R 1... R (1 ) we wsh to prove tht f R R for some (1 ). Suppose, to the cotrry, tht R 1 = 1,2,3... The for ech there s elemet R such tht 1. R, whece M R. Sce prme. Now f we me the ductve ssumpto tht sce tht 1 1 R we hve 1 1 R be bbrevto for the 1 1 R s prme the 1 1 R d R for ll R. Now R s. R, the R. Hece by complete ducto we coclude R, cotrry to our choce of d the theorem s proved. Theorem 2.4 If product of prme rght dels of terry semgroup does ot properly cot y of them, the the product s prme f d oly f t s oe of the gve dels.

3 Uo, Itersecto, Product d Drect Product of Prme Idels 1665 The proof of theorem 2.3 holds mutts mutds for dels, d for these we my drop the hypothess cocerg proper cluso theorem2.4. For, s we hve lredy oted y product of dels s coted ther tersecto, so tht the product cot cot y oe of them properly d the cluso hypothess theorem 2.4 s stsfed utomtclly. Hece we hve Theorem 2.5 A product of prme dels of terry semgroup s prme f d oly f t s oe of the gve dels. Corollry 2.6 If product of prme dels of terry semgroup s prme, the the product of the dels s just ther tersecto. The coverse of ths corollry fls, however, becuse the tersecto of the dels my be proper subset of ech of them. The tersecto of prme rght dels eed ot be prme. However, the questo whether the oempty tersecto of set of prme rght dels of terry semgroup T s prme T my be reduced to the questo whether the tersecto s prme the uo of the gve dels. Theorem 2.7 The tersecto of set of prme rght dels of terry semgroup T s prme rght del of T f d oly f t s prme rght del of the uo of the gve dels. The ecessty beg obvous, we proceed to prove the suffcecy. Let [R ] be set of prme rght dels of T; where rges over rbtrry dex set M. By hypothess, R s prme rght del of R d hece s o-empty. Let x, y, z T ; M xyz y M R. The f x R M M RL the there s some ( ) there s some R ( M ) such tht x R ; f R M such tht y R ; d f z Rj the there s some R ( M ) such tht z R. Hece x, y, z R R R. But f xyz R R R R d R, R, R re prme, whece ether x R or y R M or z R ether x R or y R or z R d ether x R or y R or z R. Therefore we hve ether x R, y R d z R or else y R, x R, z R. I ether cse, x, y, z R R R RL. But by hypothess, RL s prme j M

4 1666 Bdu. P RL, whece ether x R M or y RL or z Rj. Hece RL s prme del of T. Clerly the foregog theorem s eqully vld for dels, but for fte sets of these we hve the followg more specfc result. Theorem 2.8 The tersecto of fte set of prme dels of terry semgroup T s prme f d oly f the tersecto s oe of the gve dels. The suffcecy s trvl. To prove the ecessty, let A 1, A 2,..., A be prme dels of T, d recll tht ther tersecto must be o-empty. If 2,...,, the for ech there s elemet A such tht exctly s the proof of theorem 2.3, we my prove ductvely tht f the But 1 A 1 1 1, cotrry to the choce of, d hece coclude tht A A. Hece the supposto tht s cotrdcted, d our theorem s proved. j M A A for ll = 1, 1 1 A. Now, 1 1 A. 1 1 A A for ll = 1, 2,..., 1 Defto 2.9 The drect product of three terry semgroups R, S d T s defed to be system whose elemets re ll the ordered prs (r, s, t) wth r R, s S d t T d wth multplcto defed by (r 1, s 1, t 1 ) (r 2, s 2, t 2 ) (r 3, s 3, t 3 ) = (r 1 r 2 r 3, s 1 s 2 s 3, t 1 t 2 t 3 ). The defto my be exteded redly to the drect product of y fte set of terry semgroups. It s well ow d esly proved tht drect product of terry semgroups s terry semgroup. Our purpose troducg drect multplcto t ths pot s smply to pot out tht the drect product of prme dels of three terry semgroups eed ot be prme the drect product of the terry semgroups. A Refereces [1] Ajeyulu. A, Structure d del theory of semgroups, Thess, ANU (1980).

5 Uo, Itersecto, Product d Drect Product of Prme Idels 1667 [2] Clfford. A.H d G.P. Presto, The lgebrc theory of semgroups, vol.1, Mth.survey 7, Amer.Mth.soc, provdece, R.I, (1961). [3] Hler Brdeley Grmble, Prme dels semgroups, Thess, Uversty of Teessee (1950). [4] Jy llth.g d Srl.Y, O commuttve terry semgroups of prcpl dels, IJAER, 10 (3), (2015), [5] Kr.S, O dels terry semgroups, It.J.Mth.Ge.Sc., 18 (2005), [6] Lehmer.D.H, A terry logue of bel groups, Amer.J.Mth 54(1932), o. 2, [7] Scwrtz.S, Prme dels d mxml dels semgroups, Czech.Mth.Jour., 19 (94), (1969), [8] Shbr.M d Bsher.S, Prme dels terry semgroups, As Europe Jour. Mth., 2 (2009), [9] Soso.F.M, Idel theory terry semgroups, Mth.Jp, 10 (1965),

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