Nil Elements and Even Square Rings

Transcription

1 Interntionl Journl of Alger Vol. 7 no. - 7 HIKAI Ltd Nil Eleents nd Even Squre ings S. K. Pndey Deprtent of Mthetics Srdr Ptel University of Police Security nd Criinl Justice Dijr Jodhpur jsthn. Indi Copyright S. K. Pndey. This rticle is distriuted under the Cretive Coons Attriution License which perits unrestricted use distriution nd reproduction in ny ediu provided the originl work is properly cited. Astrct In this rticle we introduce the notion of nil eleents nd even squre rings nd provide soe exples. Nil eleents re specil type of nilpotent eleents nd possess interesting properties. Like nilpotent eleents the set of ll nil eleents in finite couttive ring fors n idel of the ring. We provide soe interesting results on couttive s well s non-couttive even squre rings. It is noticed tht if is unique non-zero nil eleent of finite couttive even squre ring then nnihiltes. In ddition it is seen tht ech nil eleent of finite couttive even squre ring does not necessrily nnihilte if it contins ore thn two nil eleents. Mthetics Suject Clssifiction : M99 Keywords: finite ring nilpotent eleent nil eleent even squre eleent even squre ring Introduction The notion of nilpotent eleents is very old nd cn e found in ech textook of odern lger (one y refer [-5]). In this rticle we introduce the notion of nil eleents nd even squre rings nd provide soe exples. It y e noted tht nil eleents re distinct fro the zero divisors nd nilpotent eleents in the sense tht every nil eleent is nilpotent however nilpotent eleent is not necessrily nil eleent. Siilrly every non-zero nil eleent is zero divisor ut zero divisor need not e nil eleent.

2 S. K. Pndey It is worth to note tht... Z the set of first non-negtive integers is couttive ring under ddition nd ultipliction odulo nd this ring contins unique non-zero nil eleent for y where y is ny non-negtive integer. But in Z the non-zero nil eleent does not nnihilte Z. However we notice tht if is finite couttive even squre ring contining unique nonzero nil eleent then nnihiltes. In ddition we consider soe exples of n even squre ring contining ore thn two nil eleents ech nil eleent of which nnihiltes. However it is noticed tht ech nil eleent of finite couttive even squre ring does not necessrily nnihilte if it contins ore thn two nil eleents. Nil Eleents nd Even Squre ings Definition. Let us cll n eleent of ring nil eleent if nd.. Fro this definition it is cler tht every nil eleent is nilpotent ut nilpotent eleent need not e nil eleent. It is lso ovious tht ech non-zero nil eleent is zero divisor however zero divisor need not e nil eleent. Definition. An eleent x x x. of ring is sid to nnihilte if Definition. Let us cll n eleent of ring n even eleent if soe. for Definition. Let e ring. Let us cll n eleent n even squre eleent if for soe. Definition 5. Let us cll ring n even ring if every eleent of is n even eleent. Definition. Let us cll ring n even squre ring if every eleent of is n even squre eleent. erk. Every even ring is n even squre ring however n even squre ring is not necessrily n even ring. Proposition. If nd re ny non-nil nilpotent eleents of finite couttive ring contining nil eleents then nd re not necessrily non-nil nilpotent eleents. efer exple. Proposition. In couttive ring the set of ll nil eleents fors n idel of.

3 Nil eleents nd even squre rings Proposition. In ring the set of ll even eleents fors n idel of. Proposition c. In couttive ring the set of ll even squre eleents fors n idel of. Corollry. If is couttive ring nd is nil eleent of then for ech is nil eleent of. Proposition. Let e finite even squre ring (couttive or noncouttive) nd let hs n nil eleents. Then hs t lest n idels provided ech nil eleent nnihiltes. Proof. Trivil. See exple 5. erk. If ech eleent of is nil eleent nd it contins n nil eleents then hs n idels. Proposition. There does not exist ny noncouttive even squre ring of order four. Proof. Let e n even squre ring of order four. If the order nd chrcteristic of re equl then it will e couttive ring. If the order nd chrcteristic of re not equl then its chrcteristic will e two. But every even squre ring of chrcteristic two is couttive. Hence there does not exist ny noncouttive even squre ring of order four. Corollry. There does not exist ny noncouttive even squre ring whose every eleent is nil eleent. Corollry. There does not exist ny noncouttive even squre ring of chrcteristic two. Proposition 5. Let e finite even squre ring of order eleent of is nilpotent. n nz then ech Proof. Trivil. See exple 5. Proposition. In finite couttive even squre ring of chrcteristic two every eleent is nil eleent. erk: There is finite couttive even squre ring of chrcteristic two t hving eleents for ech positive integer t. This ring gives finite t couttive ring of order in which x x ( is vector spce of diension t overgf ).

4 S. K. Pndey Proposition 7. Every finite couttive even squre ring of even order contins unique non-zero nil eleent provided its order nd chrcteristic re equl. Proof. Let e finite couttive even squre ring of even order. Let the order nd chrcteristic of re equl. Clerly the dditive group of is cyclic nd there is unique non-zero eleent in such tht. This iplies tht. If is n even eleent then. Therefore is nil eleent. If is not n even eleent then since is n even squre eleent therefore c for soe c. This gives c. Let then c. c c. But is the only eleent such tht nd. Hence. Proposition 8. Let e finite couttive ring nd its chrcteristic nd order re equl. If it contins unique non-zero nil eleent then is not necessrily n even squre ring. Note: Z... gives such ring for ech y ny non-negtive integer. where y is Proposition 9. Ech nil eleent of finite couttive even squre ring contining unique non-zero nil eleent nnihiltes. Proof. Let e n even squre ring. Let is ny nil eleent then. If then. If nd then. If nd nd contins unique non-zero nil eleent then. Proposition. Ech nil eleent of finite couttive even squre ring whose every eleent is n even eleent nnihiltes. Proof. Trivil. Corollry. If non-zero eleent nnihiltes ring then is ring without identity. Corollry 5. If non-zero eleent nnihiltes ring then every non-zero eleent of is zero divisor. Proposition. Let e finite couttive even squre ring contining ore thn two nil eleents then ech nil eleent of does not necessrily nnihilte.

5 Nil eleents nd even squre rings 5 Proof. See exple. EXAMPLE. One cn verify tht is finite couttive ring under trix ddition nd ultipliction odulo. Let nd c then nd c re non-zero nil eleents of. is not n even squre ring. The set of ll nil eleents of is given y N. EXAMPLE. Let. Then is finite couttive ring under trix ddition nd ultipliction odulo. It is esy to see tht c nd d re non-nil nilpotent eleents of. One cn verify tht the su nd product of ny two non-nil nilpotent eleents of re nil eleents. EXAMPLE. Let e n even positive integer nd Z : where... Z is the set of first non-negtive integers. It is ring under ddition nd ultipliction odulo. One cn esily verify tht is finite couttive even squre ring of order under trix ddition nd ultipliction odulo. Let then Z nd B is the non-zero nil eleent of. It is esy to see tht B nnihiltes.

6 S. K. Pndey EXAMPLE. Let D :. Here e the set of ll even squre eleents of Z nd n y where y is ny non-negtive integer. One y see tht D gives n even squre ring of order n under ddition nd ultipliction of trices odulo. This ring contins four nil eleents nd ech nil eleent nnihiltes D. EXAMPLE 5. Let is non-couttive even squre ring under ddition nd ultipliction of trices odulo8 hving the following properties.. hs nil eleents.. Ech nil eleent of nnihiltes.. hs t lest idels. Ech eleent of is nilpotent eleent. EXAMPLE. Let nd re ny two non-zero distinct nil eleents in couttive ring. Then 8 is finite couttive even squre ring of order 8 under the opertions defined in. Ech nil eleent of this ring does not nnihilte the ring. Acknowledgeents. The uthor is highly grteful to Prof. (Eeritus). S. Chkrvrti Deprtent of Mthetics Cochin University of Science nd Technology Cochin (Indi) for his support to construct exple. eferences [] M. Artin Alger Prentice Hll of Indi Privte Liited New Delhi. [] I. N. Herstein Topics in Alger Wiley-Indi New Delhi. [] T. W. Hungerford Alger Springer-Indi New Delhi 5.

7 Nil eleents nd even squre rings 7 [] U. M. Swy A. V. S. N. Murthy Alger-Astrct nd Modern Dorling Kindersley (Indi) Pvt. Ltd. New Delhi. [5] W. J. Wickless A First Grdute Course in Astrct Alger Mrcel Dekker Inc. New York. eceived: July ; Pulished: Jnury 5 7