The existence of a derivation in a near-ring is not known. We construct derivations on 2×2 matrix near-ring in the sense of [MW]

Nakajima, Atsushi

Okayama University Scientific Achievement Repository

Journal of the Faculty of Environmental Science and Technology, Okayama UniversityVoU, No.1, pp.l-3, January 1998Derivations on Matrix Near-RingAtsushi Nakajima·(Received October 31 , 1997)AbstractThe existence of a derivation in a near-ring is not known. We construct derivations on 2 x 2 matrixnear-ring in the sense of [MW].KEYWORDS: Near-ring, matrix near-ring, derivation1991 MATHEMATICS SUBJECT CLASSIFICATION: 16Y30.IntroductionLet R be a ring identity 1 and M 2 (R) the 2 x 2 matrix ring over R. All derivations in M2 (R) are knownnd it is used to determine the structure of a ring which has a derivation with invertible values in [BHL].)n the other hand, the notion of a derivation is useful in the theory of near-rings, and several propertiesf near-rings with derivations were given by [BM], [H] and [W]. But it is not known that there exists alerivation on matrix near-ring M2 (N), where N is a right near-ring.In this note, using a similar way as in the case ofthe matrix ring M 2 (R) over R, we construct derivationsn matrix near-ring M 2 (N).:. PreliminariesThe notion of a near-ring and related things are seen in his book [Pl. But it is not well known, so we,egin to give a definition of a near-ring.Definition 2.1. A set N with two binary operations" +" and"·" is called a near-ring if the following,roperties hold:(a) (N, +) is a group (not necessarily abelian),(b) (N,·) is a semigroup,(c) (a + b)n = an + bn for any a, b, n E N ("right distributive law").In view of (c), we call more precisely a "right near-ring". The left distributive law is defined similarlynd when this is the case, it is called a left near-ring. We also deflDe some other notions for a right near-ringV. N is called zero-symmetric if a· 0 = 0 for any a E N, and the set C(N) := {x E N Ixa = ax for any, E N} is called the center of N. A map d: N -t N is said to be a derivation ifd(a + b) = d(a) + d(b), d(ab) = d(a)b + ad(b) (a, bEN).~his is different in [P, p. 232], but it is the same as the definition oftheir papers [BM], [H] and [W].*Department of Environmental and Mathematical Sciences, Faculty of Environmental Science and Technology,Okayama University, Okayama, 700 Japan.12 1. Fac. Environ. Sci. and Tech., Okayama Univ. 3 (l) 1998In this note, we mean that a near-ring is a right near-ring and zero-symmetric. The notion of a near-ringis not well known, but there are many examples of near-rings.Example 2.2. (1) Let R be a commutative ring with identity and R[X] the set of all polynomialswith coefficients in R. Then R[X] is an additive group under the usual addition of polynomials. For f(X),g(X) E R[X], we definef(X) 0 g(X) = f(g(X)),that is, 0 is a substitution. Then (R[X], +,0) is a near-ring and zero-symmertic.(2) Let V be a vector space over a field k. We call a map f : V -+ V an affine map if f is the sum of alinear map and a constant map. Then the set Aff(V) of all affine maps is a near-ring and zero-symmerticwith pointwise addition and composition:(f + g)(v) = f(v) + g(v), (f 0 g)(v) = f(g(v)) (f,g E Aff(V), v E V).Now, we give the definition of a matrix near-ring according to J. D. P. Meldrum and A. P. J. van derWalt [MW]. Let N be a near-ring and N n the direct sum of n copies of the group (N, +). Then Nn is alsoa near-ring as usual way. We denote M(Nn) the set of all maps from (Nn,+) into itself. Then M(Nn) isa near-ring with pointwise addition and composition.Let £j = (0, ... ,0,1,0, ... ,0) (jth position is 1 and the other positions are zero). We define the followingmaps:£j : N 3 a -+ aCj = (0, ... ,0, a, 0, ... ,0) EN, (a is the jth position)7fj : N n -+ N, where 7fj(Cj) = 1, 7fj(ci) = 0 (i 'I j)r:N3x-+axENf'0 = £da 7fj : Nn -+ NnDefinition 2.3. ([MW, Definition 2.1]) The near-ring ofn x n matrices Mn(N) over N is the subnear-ring of M(Nn) generated by the set {J'0 Ia E N,I ~ i,j ~ n}.Then Mn(N) is a (right) near-ring with identity ([MW, Proposition 2.2]). In their paper [BHL], Bergen,Herstein and Lanski gave all derivations on 2 x 2 matrix ring over R. Using this method, we try to constructderivations on 2 x 2 matrix near-ring M 2 (N). By definition of fij, we see1ft : N 2 3 (a, b) -+ (xa, 0) E N 2 ,1ft : N 2 3 (a,b) -+ (O,xa) E N 2 ,1f2 : N 2 3 (a,b) -+ (xb,O) E N 21f2 : N 2 3 (a,b) -+ (O,xb) E N 2•Thus the multiplications of fij with each other are similar to the multiplication of the matrix units in ringtheory.Lemma 2.4. The multiplication of fij are follows:fijfj, = fial, fijfk, = 0, (j 'I k)for any 1 ~ i,j,k,f ~ 2 and a,b EN.3. Construction of derivationsLet a, bEN and x, y, Z, wEN. We defineA. NAKAJIMA I DerivatiollS on Matrix Near·Ring 3Ufl)' = 1ft + I~[ ,Then we have(IY )' - I by fbY12 - - 11 + 22' U~l)' = -Iff + 122' (22)' = -If; + I~r·Lemma 3.1. The following relations hold for any 1 ~ i, j, k, f. ~ 2.UiJ lIt)' = u~Y)' = Utj)'lIt + !;'jUlt)' ,UiJI%t)' = 0 = UiJ)' f%t + liJU%t)' (j f. k).Proof. We only prove the case i = j = k = f. = 1, because the other relations are proved by the similarway as in the first case. By definition, we haveU[l)' Irl + Hi Uri)' = Uft + I~f)!fl + Hi Uf? + I~n.Since M 2 (N) is a right near-ring, then the first part is expanded and the second part isH1Ufi + I~f)(s,t) = Hl((ayt,O) + (O,bys)) = Hl(ayt,bys) = (xayt,O) = g;Y(s,t).Thus Url)'!fl + Hi Uri)' = I~~Y + g;y. In this case, we see I~~y + g;y = g;y + I~~Y. This prove thecase i = j = k = f. = 1.Using this lemma, we have the followingTheorem 3.2. There exists a derivation on the matrix near-ring M2 (N).Proof. Applying the relations in Lemma 3.1 to (a, b) E N 2 , we seeUtj)' I%t + liJU%t)' = li'jU%t)' + UiJ)' f%tfor any 1 ~ i,j,k,f. ~ 2 and x,y EN. Since M2(N) is generated by the set {frl,fr2,f21,/~}and using theabove relation, we can easily see that the mapis a derivation.REFERENCES[BHL] Jeffrey Bergen, 1. N. Herstein and Charles Lanski; Derivations with invertible values, Can. J. Math.35 (1983), 300-310.[BM] E. H. Bell and G. Mason; On derivations in near-ring, Near-Rings and Near-Fields (G. Betsch, ed.),North-Holland, Amsterdam, 1987, pp. 31-35.[H] M. Hongan; On near-rings with derivation, Math. J. Okayama Univ. 32 (1990), 89-92.[MW] J. D. P. Meldrum and A. P. J. van der Walt; Matrix near-rings, Arch. Math. 41 (1986), 312-319.[P] G. Pilz; Near-Rings, Mathematics Studies, 23, 1977, North-Holland, Amsterdam.[W] Xue-Kuan Wang; Derivations in prime near-rings, Proc. Amer. Math. Soc. 121 (1994),361-366.

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