Vol. 19 (2018), No. 2, pp. 883–897 DOI: 10.18514/MMN.2018.2186
NEAR-VECTOR SPACES CONSTRUCTED FROM NEAR DOMAINS
K.-T. HOWELL AND S. P. SANON Received 18 December, 2016
Abstract. In this paper we prove some new results on near-vector spaces and near domains and give a first application of the nearring of quotients with respect to a multiplicative set, namely we construct a new class of near-vector spaces from near domains.
2010Mathematics Subject Classification: 16Y30; 12K05 Keywords: nearrings of quotients, near-vector spaces, nearrings
1. INTRODUCTION
Andr´e [1] first generalised the concept of a vector space, i.e., a linear space, to a structure comprising a bit more non-linearity, the so-called near-vector space. In [12] van der Walt showed how to construct an arbitrary finite-dimensional near- vector space, using a finite number of near-fields, all having isomorphic multiplic- ative semigroups. This construction was used in [7] to characterise all finite dimen- sional near-vector spaces overZp, forpa prime. These results were extended in [8]
to all finite dimensional near-vector spaces over arbitrary finite fields. In [6] homo- geneous and linear mappings and subspaces were investigated.
Recently, near-vector spaces have been used in several applications, including in secret sharing schemes in cryptography [4] and to construct interesting examples of families of planar nearrings [3]. In addition, they have proved interesting from a model theory perspective too [2].
In this paper we begin with some preliminary material in section 2.1. on near- vector spaces and prove some properties of isomorphisms of near-vector spaces. In section 2.2. we generalise a construction that was first considered in [6] and in sec- tion 2.3. we focus on nearrings of quotients, giving some new results that allow for alternative proofs of some of the main known results. In section 2.4. we focus on
The first author was supported in part by the South African National Research Foundation, Grant No.96056.
The second author was supported by funding from Stellenbosch University and the African Institute of Mathematical Sciences (South Africa).
c 2018 Miskolc University Press
integral nearrings and near domains and prove that if a nearringN is integral, then the nearring of quotients,NS will be integral and that ifN is a near domain andS the set of all cancellable elements ofN, thenNSwill be a nearfield. Finally, as a first application of nearrings of quotients, in section 2.5. we use the results from section 2.4. to construct a new class of near-vector spaces from near domains.
2. RESULTS
2.1. Preliminary considerations
Definition 1([1]). A pair.V; A/is called anear-vector spaceif:
(1) .V;C/is a group andAis a set of endomorphisms ofV; (2) Acontains the endomorphisms0,idand id;
(3) ADAn f0gis a subgroup of the group Aut.V /;
(4) Aacts fixed point free (fpf) onV, i.e., forx2V; ˛; ˇ2A,x˛Dxˇimplies thatxD0or˛Dˇ;
(5) the quasi-kernelQ.V /ofV, generates V as a group. Here, Q.V /D fx 2 Vj 8˛; ˇ2A;9 2Asuch thatx˛CxˇDxg.
We sometimes refer to V as a near-vector space over A. The elements of V are called vectorsand the members of Ascalars. The action of Aon V is called scalar multiplication. Note that id 2Aimplies that .V;C/ is an abelian group.
Also, the dimension of the near-vector space, dim.V /, is uniquely determined by the cardinality of an independent generating set forQ.V /. See [1] for further details.
In [12] van der Walt proved the following theorem,
Theorem 1([12] Theorem 3.4, p.301). LetV be a group and letAWDD[ f0g, where D is a fix point free group of automorphisms ofV. Then .V; A/ is a finite dimensional near-vector space if and only if there exists a finite number of nearfields, F1; F2; : : : ; Fn, semigroup isomorphisms i WA!Fi and a group isomorphism˚W V !F1˚F2˚ ˚Fnsuch that if
˚.v/D.x1; x2; ; xn/; .xi2Fi/ then
˚.v˛/D.x1 1.˛/; x2 2.˛/; ; xn n.˛//;
for allv2V and˛2A.
According to this theorem we can specify a finite dimensional near-vector space by taking nnearfields F1; F2; : : : ; Fn for which there are semigroup isomorphisms
#ij W.Fj;/!.Fi;/with#ij#j kD#i k for1i; j; kn. We can then takeV WD F1˚F2˚ ˚Fnas the additive group of the near-vector space and any one of the semigroups (Fio,) as the semigroup of endomorphisms by defining
.x1; x2; : : : ; xn/˛WD.x1#1io.˛/; x2#2io.˛/; ; xn#nio.˛//;
for allxj 2Fj and all˛2Fio.
Definition 2([8] Definition 3.2, p.4). We say that two near-vector spaces.V1; A1/ and .V2; A2/ are isomorphic (written .V1; A1/Š.V2; A2/ if there are group iso- morphismsW.V1;C/!.V2;C/andW.A1;/!.A2;/such that.x˛/D.x/.˛/
for allx2V1and˛2A1.
We denote this pair by.; /. Isomorphisms map quasi-kernels to quasi-kernels:
Proposition 1. If the near-vector spaces.V1; A1/and.V2; A2/are isomorphic, thenQ.V1/is mapped toQ.V2/:
Proof. Letv2Q.V1/. IfvD0, then.v/2Q.V2/, so suppose thatv¤0. Let
˛; ˇ2A2, then
.v/˛C.v/ˇD.v/.˛1/C.v/.ˇ1/for some˛1; ˇ12A1; D.v˛1/C.vˇ1/
D.v˛1Cvˇ1/
D.v1/for some12A1; D.v/.1/:
Thus.v/2Q.V2/.
It is not difficult to show that
Lemma 1. If.; /is an isomorphism of.V1; A1/onto.V2; A2/, thenis uniquely determined on any basis of.V1; A1/.
In [1], the concept of regularity is introduced as a central notion. A near-vector space is regular if any two vectors of Q.V /nf0g are compatible, i.e. if for any two vectors uandv ofQ.V /there exists a 2Anf0gsuch that uCv2Q.V /.
Every near-vector space can be uniquely decomposed into a direct sum of regular near-vector spacesVj (j 2J) ([1], Theorem 4.13, p.12) and there is a unique direct decomposition into maximal regular near-vector spaces, called thecanonical decom- positionofV. Thus the theory of near-vector spaces is largely reduced to the theory of regular near-vector spaces.
Theorem 2. If the near-vector spaces.V1; A1/and.V2; A2/are isomorphic and .V1; A1/is regular,.V2; A2/will be regular too, i.e. isomorphisms preserve regular- ity.
Proof. By definition there exist group isomorphisms W.V1;C/!.V2;C/ and W.A1;/!.A2;/such that.x˛/D.x/.˛/for allx2V1and˛2A1. Let w1; w22Q.V2/nf0g. Then there existv1; v22Q.V1/(by Proposition1), such that w1D.v1/ andw2D.v2/. Since V1 is regular there exists a2A1 such that v1Cv22Q.V1/. Thus
.v1Cv2/D.v1/C.v2/./
Dw1Cw2./with./¤0:
Thus.V2; A2/is regular.
We also have that
Lemma 2. Let .V1; A1/ and .V2; A2/ be near-vector spaces and .; / an iso- morphism. If V1 DL
j2JVj is the canonical direct decomposition of V1, then V2DL
j2J.Vj/is the canonical direct decomposition ofV2. The proof is similar to that of Lemma 5.5, p.2537 in [6].
2.2. Near-vector spaces of the formFnwhereF is a nearfield
In [6] van der Walt’s characterisation was used to discuss several constructions of near-vector spaces. In particular, the case where we letV DFn,n2N,F a nearfield and we take all the isomorphisms to be identical, so that
.x1; x2; : : : ; xn/˛D.x1˛; x2˛; : : : ; xn˛/
was considered.
This is the case closest to the normal vector space setting. Let us denote it by .V; F /. In fact, whenF is a field,.V; F /is a vector space. We will denote the set of all distributive elements ofF byFd (as in [10]), i.e.
Fd D fd 2Fjd.xCy/DdxCdy8x; y2Fg:
It is not difficult to check thatFdis a subfield ofF. Note that0; 12Fd. It was shown in [6] thatQ.V /D [Vi, withVi D.d1; d2; : : : ; 1; diC1; : : : ; dn/F, with1in position i anddi 2Fd,i 2 f1; : : : ; ngand that.V; F /is a regular near-vector space (Lemma 3.5., p.2531).
We now generalise,
Theorem 3. Let F be a near-field andV DFn, n2Nbe a near-vector space with the scalar multiplication defined by
.x1; : : : ; xn/˛D.x1.˛/; : : : ; xn.˛//
whereis an automorphism of.F;/and˛2F:Then
Q.V /D f.di/j2F; di 2Fd for alli2Ig; whereID f1; 2; : : : ; ng:
Proof. Letdi2Fd fori2 f1; : : : ; ngand˛; ˇ2F. We have .di/˛C.di/ˇD.di.˛/Cdi.ˇ//
D.di..˛/C.ˇ///sincedi is distributive, D.di. //withD 1..˛/C.ˇ//, D.di/:
Hence .di/2Q.V /. Since Q.V / is closed under scalar multiplication we have Q.V / f.di/j2F; di 2Fd for alli 2Ig: Now suppose that .xi/2Q.V /. If .xi/D0 then .xi/2 f.di/j2F; di 2Fd for alli 2Ig. Suppose that .xi/¤0.
Then there isi02 f1; : : : ; ngsuch thatxi0¤0:LetdiDxixi 1
0 fori2 f1; : : : ; ng:Then .di/D.xi/ 1.xi 1
0 /. Since.xi/2Q.V /andQ.V /is closed under scalar multiplic- ation,.di/2Q.V /. Then for all˛; ˇ2F there is 2F such that.di/˛C.di/ˇD .di/: It follows thatdi.˛/Cdi.ˇ/Ddi. / for alli 2 f1 : : : ; ng:Since is an automorphism, there are˛1; ˇ1 2F such that ˛D 1.˛1/andˇD 1.ˇ1/. So di˛1Cdiˇ1Ddi. /for alli2 f1 : : : ; ng:Butdi0D1. So˛1Cˇ1D. /:Hence di˛1Cdiˇ1Ddi.˛1Cˇ1/, and this is verified for all ˛1; ˇ12F because is an automorphism of.F;/. Thereforedi is distributive andQ.V /D f.di/j2F; di 2
Fd for alli2Ig:
In fact, if we takeV DFn,n2NwithF a near-field and denote the near-vector space in the above theorem by.V; F0/, we can show that
Proposition 2. The near-vector spaces.V; F /and.V; F0/are isomorphic for the same nearfieldF.
Proof. Using Definition2, consider the identity isomorphism andW.F;/! .F;/the mapping.˛/D 1.˛/. Sinceis an isomorphism, 1exists andis an isomorphism. Thus.x˛/D.x/.˛/for allx2V and˛2F.
Finally, since by Theorem2isomorphisms preserve regularity, we have that Lemma 3. .V; F0/is a regular near-vector space.
We end off the section with an example,
Example1. Consider the field (GF .32/,C,) with
GF .32/WD f0; 1; 2; ; 1C; 2C; 2; 1C2; 2C2g; where is a zero ofx2C12Z3Œx.
The operations onGF .32/can be defined as follows:
C W.aCb /C.cCd /D.aCc/mod3 C..bCd /mod3/;
0 1 2 1C 2C 2 1C2 2C2
0 0 0 0 0 0 0 0 0 0
1 0 1 2 1C 2C 2 1C2 2C2 2 0 2 1 2 2C2 1C2 2C 1C 0 2 2 2C 2C2 1 1C 1C2 1C 0 1C 2C2 2C 2 1 1C2 2 2C 0 2C 1C2 2C2 1 1C 2 2 2 0 2 1 1C2 1C 2 2C2 2C 1C2 0 1C2 2C 1C 2 2 2C2 1 2C2 0 2C2 1C 1C2 2 2C 1 2
In [11], p.257, it is observed that (GF .32/,C,ı), with xıyWD
xy ifyis a square in (GF .32/,C,)
x3y otherwise
is a (right) nearfield, but not a field.
ı 0 1 2 1C 2C 2 1C2 2C2
0 0 0 0 0 0 0 0 0 0
1 0 1 2 1C 2C 2 1C2 2C2 2 0 2 1 2 2C2 1C2 2C 1C 0 2 2 1C2 1C 1 2C2 2C 1C 0 1C 2C2 2C 2 2 1C2 1 2C 0 2C 1C2 2C2 2 1C 1 2
2 0 2 1 2C 2C2 2 1C 1C2 1C2 0 1C2 2C 1C 2 1 2C2 2 2C2 0 2C2 1C 1C2 1 2C 2 2 It is not difficult to see that the the distributive elements of (GF .32/, C, ı), de- noted by (GF .32/, C, ı/d are the elements 0; 1; 2. Consider the near-field F D .GF .32;C;ı/, putV DF3with˛2F acting as an endomorphism ofV by defining .x1; x2; x3/˛D.x1˛3; x2˛3; x3˛3/:Thus we have thatQ.V /D f.d1; d2; d3/j2 F; di 2 f0; 1; 2ggand this near-vector space is regular.
2.3. Nearrings of quotients
The concept of nearrings of quotients was first defined by Maxson [9] and he stated conditions for a nearring to have a nearring of quotients. Graves and Malone [5] subsequently generalised this to the case of nearrings of quotients with respect to a multiplicative set.
We consider the case whereN is a non-commutative nearring and begin with the basic results as found in [5,11]. Corollary1and2are new results that we use to give alternative proofs to the known results of Theorem4and Theorem5(See [11]). For more on nearrings we refer the reader to [10,11].
Definition 3([11] Definition 6.3, p.26). LetNbe a nearring andSa subsemigroup of.N;/. A nearringNsis called a nearring of right quotients ofN with respect toS if
(1) Nsis a nearring with identity,
(2) N is embeddable inNs, by a homomorphism, sayh, (3) 8s2S,h.s/is invertible in.Ns;/,
(4) for allq2Ns, there existss2S andn2N such thatqDh.n/h.s/ 1: We can also define a nearring of left quotients ofN with respect toS, which has the same definition as above, except property4which becomes :
4’. for allq2Ns, there existss2S andn2N such thatqDh.s/ 1h.n/:
It is not difficult to see that any element ofNscan be written ash.n/h.s/ 1, fors2S andn2N.
Definition 4([11] Definition 6.4, p.26). LetNbe a nearring andSa subsemigroup of .N;/. N is said to fulfill the right Ore condition with respect to S, if for all .n; s/2NS, there exists.n1; s1/2NS such that ns1Dsn1. Likewise, N is said to fulfill the left Ore condition with respect toS if for all.n; s/2NS, there exists.n1; s1/2NS such thats1nDn1s.
IfN is a nearring andS, a subsemigroup of .N;/ and if for alls2S, s is can- cellable (both sides) andN satisfies the right Ore condition with respect toS, then the relationdefined onNSby
.n; s/.n0; s0/if9.n1; s1/2N such thatss1Ds0n1implies thatns1Dn0n1
is an equivalence relation. Moreover, on the equivalence class of .n; s/2NS, denoted by ns, we define the operations ”C” and ”” by
n sCn0
s0 Dns1Cn0n1
ss1
and n
sn0
s0 Dnn2
s0s2
;
where.n1; s1/2NSand.n2; s2/2NSfulfills0n1Dss1andn0s2Dsn2. These operations are well-defined and.NsDNS=;C;/is a nearring of right quotients ofN with respect toS.
We prove a new result:
Corollary 1. LetNsbe a nearring of right quotients ofN with respect toS. Then h.n/h.s/ 1Dh.n0/h.s0/ 1if and only if.n; s/.n0; s0/for all.n; s/and.n0; s0/in NS.
Proof. Let .n; s/; .n0; s0/2NS such that .n; s/ .n0; s0/. Then there exists .n1; s1/2NS such thatss1Ds0n1impliesns1Dn0n1. It follows that
h.s/h.s1/Dh.s0/h.n1/ h.n/h.s1/Dh.n0/h.n1/:
So, since for alls2S,h.s/is invertible inNs, h.n1/Dh.s0/ 1h.s/h.s1/
h.n/Dh.n0/h.n1/h.s1/ 1: Therefore
h.n/Dh.n0/h.s0/ 1h.s/h.s1/h.s1/ 1 Dh.n0/h.s0/ 1h.s/:
Thush.n/h.s/ 1Dh.n0/h.s0/ 1:
Now we show the converse. Let .n; s/; .n0; s0/2NS such that h.n/h.s/ 1D h.n0/h.s0/ 1: Then there exists .n1; s1/ 2 N S such that h.s0/ 1h.s/ D h.n1/h.s1/ 1, by property4in the definition of a nearring of right quotients with re- spect toS. Soh.ss1/Dh.s0n1/. Sincehis a monomorphism,ss1Ds0n1. Also, since h.s/h.s1/ D h.s0/h.n1/, h.s0/ 1h.s/h.s1/ D h.n1/. So h.s0/ 1 D h.n1/h.s1/ 1h.s/ 1. Using the fact that h.n/h.s/ 1 D h.n0/h.s0/ 1, we get h.n/h.s/ 1 Dh.n0/h.n1/h.s1/ 1h.s/ 1: So h.n/h.s1/Dh.n0/h.n1/. Therefore h.ns1/Dh.n0n1/ and ns1 Dn0n1, since h is a monomorphism. Thus .n; s/
.n0; s0/.
The use of this corollary allows us to give an alternate proof to that found in [11], of the following theorem:
Theorem 4([11] Theorem 1.65, p.27). LetNbe a nearring andSa subsemigroup of.N;/.N has a nearring of right quotients with respect toSis equivalent to
(1) for alls2S,sis cancellable (on both sides), (2) N satisfies the right Ore condition with respect toS.
Proof. LetS be a subsemigroup of.N;/. Suppose that N has a nearringNs of right quotients with respect toS. SoN is embeddable inNs, by a homomorphismh.
SinceSis a subsemigroup,S¤¿:Lets2Sandn; n02N, such thatn0sDns. Then h.n0/h.s/Dh.n/h.s/:Since h.s/is invertible in.Ns;/, h.n0/Dh.n/. It follows thatn0Dn, sincehis a monomorphism. Also ifsn0Dsn, we have n0Dn. Thus for all s2S, s is cancellable on both sides. Now, let n2N and s2S. Then h.s/ 1h.n/2Ns. So by property4of Definition3, there exists.s1; n1/2SN, such thath.s/ 1h.n/Dh.n1/h.s1/ 1:Henceh.ns1/Dh.sn1/, andns1Dsn1. Therefore N satisfies the right Ore condition with respect toS.
To show the converse, we suppose thatSis not empty, for alls2S,sis cancellable (on both sides) and thatN satisfies the right Ore condition with respect toS. So from the discussion following Definition4there exists a nearring of right quotients ofN
with respect toS, namely.NS=;C;/.
Remark 1. We note that there is a printing error in the statement of the above theorem in [11](Theorem 1.65, p.27), the left Ore condition should be replaced with the right Ore condition.
Example2. Let us consider the nearring.M.R/;C;ı/. An elementf 2M.R/is cancellable if and only iff is bijective. In fact iff is bijective, thenf is cancellable.
So let us supposef is cancellable. That implies that for allg; g02M.R/,f ıgD f ıg0 impliesgDg0, alsogıf Dg0ıf implies gDg0. It is not difficult to see thatf ıgDf ıg0impliesgDg0, implies thatf is injective. Also ifgıf Dg0ıf impliesgDg0, thenf is surjective.
So now let us considerN D.RŒx;C;ı/M.R/, the set of polynomials defined fromRC, andSthe set of all monomials inN with co-domain inRC(in other words monomials of the formaxn, witha>0andn2N). Since each monomial is defined on RC, every element of S is bijective, then cancellable. Also the composition of two monomials is a monomial. ThusS is a subsemigroup of.N;ı/. Moreover for .f; g/2NS,f ıid Dgı.g 1ıf /. But.g 1ıf; id /2NS, soN satisfies the right Ore condition with respect toS. HenceN has a nearring of right quotients with respect toS.
The quotient is a set.Ns;C;ı/, whereNsis the set of all summations of all power functionsf .x/Dx˛, with˛2QCandf defined fromRCtoR.
We prove a new corollary:
Corollary 2. LetN be a nearring andS a subsemigroup of .N;/. If N has a nearring of right quotients with respect toS,Ns, then there exists.n1; s1/2NS such that h.n/h.s/ 1Ch.n0/h.s0/ 1Dh.ns1Cn0n1/h.ss1/ 1, wheress1Ds0n1
andhis the embedding homomorphism fromN toNs. Also, there exists.n2; s2/2 NSsuch thath.n/h.s/ 1h.n0/h.s0/ 1Dh.nn2/h.s0s2/ 1, wheren0s2Dsn2.
Proof. Since N has a nearring of right quotients with respect to S, N satisfies the right Ore condition with respect toS. So there exists.n1; s1/2NS such that ss1Ds0n1. Thenh.s0/ 1Dh.n1/h.s1/ 1h.s/ 1. We have
h.ns1Cn0n1/h.ss1/ 1Dh.n/h.s1/h.s1/ 1h.s/ 1Ch.n0/h.n1/h.s1/ 1h.s/ 1: Henceh.ns1Cn0n1/h.ss1/ 1Dh.n/h.s/ 1Ch.n0/h.s0/ 1:
Also, since.n0; s/2NS, there exists.n2; s2/2NS such thatn0s2Dsn2, by the right Ore condition. Soh.n2/Dh.s/ 1h.n0/h.s2/.
We have
h.nn2/h.s0s2/ 1Dh.n/h.n2/h.s2/ 1h.s0/ 1
Dh.n/h.s/ 1h.n0/h.s2/h.s2/ 1h.s0/ 1:
Henceh.nn2/h.s0s2/ 1Dh.n/h.s/ 1h.n0/h.s0/ 1. We will use the above corollary to give an alternate proof of the following result found in [11]:
Theorem 5([11] Theorem 1.66, p.28). LetN be a nearring. If Ns andNs0 are nearrings of right quotients with respect toS, then
NsŠNs0:
Proof. SinceNs andNs0are nearrings of right quotients, there exist monomorph- ismshandh0, fromN toNsandNtoNs0, respectively. Let us define the mappingf
by
f WNs!Ns0
h.n/h.s/ 17!h0.n/h0.s/ 1:
The mapping f is well-defined. To show this suppose that h.n/h.s/ 1 D h.n0/h.s0/ 1: Then by Corollary 1 .n; s/.n0; s0/. Also by the same Corollary 1h0.n/h0.s/ 1Dh0.n0/h0.s0/ 1:So f .h.n/h.s/ 1/Df .h.n0/h.s0/ 1/:Thusf is well-defined. Leth.n/h.s/ 1; h.n0/h.s0/ 12Ns. Then by Corollary2we have
f .h.n/h.s/ 1Ch.n0/h.s0/ 1/Df .h.ns1Cn0n1/h.ss1/ 1/ .n1; s1/2NS fulfillings0n1Dss1
Dh0.ns1Cn0n1/h0.ss1/ 1
Dh0.n/h0.s/ 1Ch0.n0/h0.s0/ 1sinces0n1Dss1
Df .h.n/h.s/ 1/Cf .h.n0/h.s0/ 1/:
Also by Corollary2again, we have
f .h.n/h.s/ 1h.n0/h.s0/ 1/Df .h.nn2/h.s0s2/ 1/;where.n2; s2/2NS fulfillsn0s2Dsn2
Dh0.nn2/h0.s0s2/ 1
Dh0.n/h0.s/ 1h0.n0/h0.s0/ 1;sincen0s2Dsn2
Df .h.n/h.s/ 1/f .h.n0/h.s0/ 1/:
Thus f is a homomorphism. Now, let h.n/h.s/ 1; h.n0/h.s0/ 1 2Ns such that h0.n/h0.s/ 1Dh0.n0/h0.s0/ 1:Then.s; n/.s0; n0/. Soh.n/h.s/ 1Dh.n0/h.s0/ 1: Hencef is injective. Leth0.n/h0.s/ 12Ns0. For.n; s/2NS,h.n/h.s/ 12Ns
andf .h.n/h.s/ 1/Dh0.n/h0.s/ 1:Hencef is a surjection. Thereforef is a bijec-
tion. ThusNsŠNs0:
Theorem5allow us to speak of the nearring of right quotients,Ns, with respect to S, for a particular nearring.
2.4. Integral nearrings and near domains
Recall that a nearring.N;C;/is said to beintegralif it has no non-zero divisors of zero and
Definition 5. ([5] Definition 1.4, p.34) A near domain is a nearringNthat satisfies the right Ore condition and the left cancellation law.
Then we have the following:
Theorem 6. LetN be a nearring and S a subsemigroup of.N;/. Suppose that N has a nearring of right quotients with respect toS,Ns. IfN is integral, thenNs
is integral.
Proof. Leth.n/h.s/ 1; h.n0/h.s0/ 12Ns withhthe monomorphism fromN to Ns. We have
h.n/h.s/ 1h.n0/h.s0/ 1Dh.nn2/h.s0s2/ 1; withsn2Dn0s2for.n2; s2/2NS:
Suppose thath.nn2/h.s0s2/D0. Thenh.nn2/h.s0s2/ 1Dh.0/h.t / 1, for somet2 S. This implies that.nn2; s0s2/.0; t /. It follows that there exits.n1; s1/2NS such thatnn2s2D0. s2¤0becauseh.s2/is invertible. Hencenn2D0, sinceN is integral. Moreover, sinceNis integral,nD0orn2D0.n2D0implies thatn0s2D0 and son0D0. ThereforenD0orn0D0. Thush.n/.s/ 1D0orh.n0/h.s0/ 1D0.
ThusNsis integral.
Proposition 3. LetN be a near domain. Then (1) 0nDn0D0, for alln2N
(2) N is integral
(3) N satisfies the right cancellation law.
Proof. LetN be a near domain andn; n1; n22N.
(1) It is straightforward to show that0nD0. We have.n0/.n0/Dn.0n/0Dn0.
So.n0/.n0/0D.n0/0. Using the left cancellation law we haven0D0.
(2) Supposen1n2D0. Ifn1¤0we have n1n2Dn10. It follows thatn2D0 from the left cancellation law.
(3) Ifn1nDn2n, withn¤0, then.n1 n2/nD0. Hence by 2.n1Dn2. Corollary 3. LetN be a near domain. LetSbe the set of all cancellable elements ofN. ThenNsis a nearfield.
Proof.
SinceN is a near domain, every non-zero element is cancellable. SoS DN f0g. We know from Definition3thath.s/is invertible for everys2S. Letq2Ns. Then there exist.n; s/2NSsuch thatqDh.n/h.s/ 1:Supposeq¤0. Thenn¤0. So n2S. Henceh.n/is invertible. Thereforeh.n/h.s/ 1is invertible and its inverse is
h.s/h.n/ 1.
2.5. An application to Near-vector spaces
In this last section we use the results of the previous sections to give a first applic- ation of the theory of nearrings of quotients. We construct a new class of near-vector spaces from near domains and completely describe the quasi-kernel.
IfN is a nearfield andSDN f0g, thenNs is a nearfield. In fact, we can show thatN'Ns. From the definition of a nearring of quotients we know that there is an embedding maphdefined by
hWN!Ns
n7! n 1
with1the multiplicative identity ofN. We just have to show thathis surjective. Let .n; s/2NS. We have thatns Dns11. Soh.ns 1/Dns. Hencehis an isomorphism and soN'Ns. ThereforeqDh.n/2Nsis distributive if and only ifnis distributive.
Thus ifN is a nearfield then
Ns˚: : :˚Ns'N˚: : :˚N
and the study of constructions of the formN˚: : :˚N has been discussed in [6].
We now look at the case whereN is a near domain, not necessarily a nearfield.
Let us considerNs (with identity1) withS the set of all cancellable elements. We takeV DNsL
: : :L
Nswith the scalar multiplication defined for.x1; : : : ; xn/2V and˛2Nsby
.x1; : : : ; xn/˛D.x1˛; : : : ; xn˛/:
We now look at the quasi-kernel Q.V /. We know from [6] that the quasi-kernel Q.V /DV1[: : :[Vn, where Vi D f.d1; : : : ; di 1; 1; diC1; : : : ; dn/Nsjdi 2Ns dg, with Ns d representing the distributive elements of Ns. Thus in order to describe Q.V /, we need to find the elements ofNs d.
Theorem 7. Ns d D fh.n/h.s/ 1jif9a; b; c2N; such that saCsbDsc; then naCnbDnc fors2S; n2Ng
Proof.
Let q Dh.n/h.s/ 1; q1 Dh.n1/h.s1/ 1; q2Dh.n2/h.s2/ 1 2Ns. First suppose thatqis distributive and that there area; b; c2N such thatsaCsbDsc. We prove thatnaCnbDnc. We have
h.n/h.s/ 1.h.saCsb//Dh.n/h.s/ 1.h.sc//:
But
h.n/h.s/ 1.h.saCsb//Dh.n/h.s/ 1.h.s/h.a/Ch.s/h.b//
Dh.n/h.a/Ch.n/h.b/; sinceh.n/h.s/ 1is distributive.
Dh.naCnb/;
and
h.n/h.s/ 1.h.sc//Dh.nc/:
Hence, since h is injective, naCnb Dnc. To show the converse, suppose that if there exista; b; c2N such thatsaCsbDsc, thennaCnbDnc. We have to prove
thath.n/h.s/ 1is distributive inNs. So we have to showq.q1Cq2/Dqq1Cqq2: We have
q.q1Cq2/Dh.n/h.s/ 1h.n1sCn2n/h.s1s/ 1wheres1sDs2n Dh.nn1/h.s1ss1/ 1wheresn1D.n1sCn2n/s1; and
qq1Cqq2Dh.nn0/h.s1s0/ 1Ch.nn10/h.s2s10/ 1 wheresn0Dn1s0; sn10Dn2s10
Dh.nn0s20Cnn10n20/h.s1s0s20/ 1wheres1s0s20Ds2s10n20: To show thath.nn1/h.s1ss1/ 1Dh.nn0s20Cnn10n20/h.s1s0s20/ 1, we have to find.n; s/2NSsuch thats1ss1sDs1s0s20nimpliesnn1sD.nn0s20C nn10n20/n. SinceNsis a nearring of right quotients with respect toS, we have the right Ore condition with respect toS. Thus, sinces1ss12N; s1s0s20 2S, there exist.n; s/2NSsuch thats1ss1sDs1s0s20n. Buts1ss1sDs1s0s20n impliess2ns1sDs2s10n20n, becauses1s0s20Ds2s10n20 ands1sDs2n. So we getss1sDs0s20nandns1sDs10n20n, sinces1 ands2are cancellable.
Also we have
sn0s20nDn1s0s20n; sincesn0Dn1s0 Dn1ss1s; sincess1sDs0s20n and
sn10n20nDn2s10n20n; sincesn10 Dn2s10 Dn2ns1s; sincens1sDs10n20n: So we get
sn0s20nCsn10n20nDn1ss1sCn2ns1s D.n1sCn2n/s1s
Dsn1s; sincesn1D.n1sCn2n/s1:
Hences1ss1sDs1s0s20nimpliessn0s20nCsn10n20nDsn1s. If we take aDn0s20n; bDn10n20n; cDn1s, from our assumption we havenaCnbDnc.
Sonn0s20nCnn10n20nDnn1s. Thereforeh.nn1/h.s1ss1/ 1Dh.nn0s20C nn10n20/h.s1s0s20/ 1. Thush.n/h.s/ 1is distributive.
Using Theorem7we can describe the quasi-kernelQ.V /of the near vector space V defined above just by considering the elements ofN. So we can construct a near vector space over a nearfield from a near domain.
In closing we can now describeQ.V /for the above near-vector space.
Corollary 4. Let us consider the near vector spaceV defined above, and let N1D f.n; s/2NNjif9a; b; c 2N; such that saCsbDsc; thennaCnbD nc fors2S; n2Ng. Then we have
QDV1[: : :[Vn; where
ViD f.d1; : : : ; di 1; 1; diC1; : : : ; dn/Nsjdi Dh.ni/h.si/ 1; .n; s/2N1g: Moreover, by Lemma 3.5 [6], this near-vector space is regular.
ACKNOWLEDGEMENT
The authors express their gratitude to the NRF(South Africa), Stellenbosch Uni- versity and AIMS (South Africa) for funding that made this research possible.
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Authors’ addresses
K.-T. Howell
Stellenbosch University, Department of Mathematical Sciences, Stellenbosch, South Africa E-mail address:kthowell@sun.ac.za
S. P. Sanon
Stellenbosch University, Department of Mathematical Sciences, Stellenbosch, South Africa E-mail address:sogos@sun.ac.za
