Mathematical Analysis and Applications Editors: S. Nanda and a.p. Rajasekhar Copyright 2004. Narosa Publishing House. New Delhi. India Ring with Involution Introduced by a New Product Department of Mathmatics. Indian Institute of Technology Kharagpur Kharagpur- 721302. India Let R be any associative ring with involution. The new product <> is introduced in R. This product is not associative. We shall Si,LY U as a left-o-ideal of IUf (i) U is a additive subgroup and (ii)rou E U Vr E R,u E U. Then every ideal of R is also a left-o- ideal, but the converse is not true always. Obviously left-o-ideals satisfy the property that intersection of any two left-<>-ideals is a left-o--ideal. We denote S = {x E Rlx = x} and I< = {x E Rlx = -'Ix} for symmetric and skewsymmetric elements of R. If R is a simple hng of characteristic: not 2, 2R is an ideal of R and so must be R. Hence the relation 2r = (r +r' + (r - r' gives R = S + I<, S n I< = O. The commutators of R are defined by lx, yj = xy - yx for all x, y E R. The centre of R is denoted by Z(R) or simply by Z. Therefore for any y E 5, (.TOy)' = xoy Le.,.TOy E S i.e., RoS ~ 5. Since S is additive subgroup, 5 is a lcft-o-ideal of R. Similar manner we can show that I< is also l' left-o-idf'al of R.
Example 2.2 Let = {x E RIRox = OJ. As is additive subgroup, from definition of, it is left-o-ideal of R. For x, y E and r E R, r 0 x = 0 gives rx = -xr- and r 0 y = 0 gives ry = -yr-. Now rxy = (rx)y = (-xr )y = -x(rft ) = -xc -yr) = xyr that is Ir,xy] = 0 'Vx, Y E and r E R. This implies 2 ~ Z. Let R be a prime ring and x E, r, s E R. Then rs 0 x = 0 gives By using the property rx = -xr 'Vx E L, r E R, equation (1) reduces to (rs + sr)x = O. Again putting r = rt, t E R we get Ir, sjtx = 0 i.e., [R, R)R = O. Since R is prime ring, either R is commutative or = O. Example 2.3 If we denote R 0 R the additive subgroup generated by all ri 0 Si; ri, Si E R, then R 0 R is a left-o-ideal. Similarly R 0 (R 0 R), R 0 (R 0 (R 0 R)),... are allleft-o-ideals of R. TI.eorem 3.1 Let R be a prime ring. If a E R commutes with all x oy; x,y E R then [a,x](xoy) = 0 and a E Z. Proof. fa, xoy] = 0 gives a(xy+yx ) = (xy+yx )a. Now putting y = xy we get a(xxy + xyx ) = (xxy + xyx )a i.e., la, x](x 0 y) = O. Again we put y = ya. Then o [a, x](xya + yax ) la, x](xy + yx )a - [a, xjy(x u - ax ) [a, x]y[a, x ] = la, xlrla, Since R is prime ring, either a commutes with ;1; or x V x E R. We sl.'t 1'1 = {x E R Ira, :r] = O} and 72 = {x E R I[a,:r'l = OJ. Then 1'] :ti1d 72 are subring of R alld 1'1 U 1'2 =.: R. Since It gwup can not be ullion of two subgroups. therefore either 1'1 = R or 1'2 = R. In both cases a E Z. Theorem 3.2 Let R be a prime ring. If ao(xoy) = 0 ";''f,y E R then a E Z, P7'Oof. Putting y = xy in the given condition we get ao(:roxy) = 0 which reduccs to [a.. r)(xy ) = O. Hence by Theorem 3.1. it. follows the thcon'hi. Theorem 3.3 If L is a left-o-ideai of R sur:h t.hat L 2 =.: L t.hen L should be a Lie ideal of R. x ].
})rooj. Since L is lcfh>-ickal of R, roy, r' 0 x E L Ii x, Y E L, r E R. Again as L2 == L tljcrefore (7' 0 y)x - y(l" 0 x) == [1', yxj E L, Ii x, Y E L,r E R. Therefc,re [R, CI s;;: L. Siucc L 2 == L, [R, L] S;; L and theorem is proved. Now by theorem.1.3 and [2,Theorcm 1.2 ], the following corollary is straight forward Corollary 3.4 Let R be a simple ring of characteristic t= 2. If L is a li:ft-o-ideal such that L 2 == L t.hen either L == R or L S;; Z. Theorem 3.5 Let R be a prime ring with involution. L f- 0 is a subring as well as left-o-iueal of R then L coutains an ideal generated by all yo:!: ---2y.T, for all :r,!j E L, otherwise L will he commutative with trivial involution uu L. }J/'()of. Fur all.', y E L alld r E R, (7' 0 (y 0 x -- 2:.cy)}s' - (y 0 x - 2xy)rs ' r 0 (y C:l: - 2xy)s' - (y 0 x - 2xy)s'r' E L. Tllus L coutain an idr:al < yo/' -- 2yx >x, V Y E L. Now assume that yo:); - 2t11 == 0 i.e., xy' == yx V x, Y E L. Now putting y =-.: zy, z E L, we get :rtzy)* == (zy)x which gives [V, z]x == Ox;V y, z E L. Sillce L is Icft-o-ideal, we put x == l' 0 x, l' E R. Then it gives IY, z](,. 0 x) == 0 which implies [y, z]rx --"-0 Ii x, y, z E L, l' E R. Now since n is prillle rill~r, ('it/wr L == 0 or L is commutative. If L is commutative we get from xy' == yx by putting x == r' ox, r E R, :rr(y-y')==o Vx,yELandrER. Again by using primeness of n, we get y == y' Ii y E L. It follows the theorem. Theorem 3.6 Let R be a ring with involution and L is its a left-o-ideal. Theil L contains ideals generated by all x 0 y - y' 0 x, x, Y ELand [.c, y] + [y', x], x, Y E L; otherwise every clements of L will be normal. Proof. The i(knlity
Therefore, R(x 0 y - y' 0 x)r <; L If x, Y E L. From here we can write R{(:roy-y'ox)+(yox-x'oy)}R~' which gives R([x',yJ+[Y',xj)R <; L, If x, Y E L. x 0 y - y' 0 x = 0 and [:r', V] + [y', xl = 0 both cases give xx' = x'x i.e., every elements of L is normal. Theonml 3.7 Let R be a simple ring. Tllen Sand K do not contain any left-o-idta]s of R except themselves. Pmof. Let L is any left-o-ideal of R. Therefore L is a Jordan ideal of S. Now from 12,Theorem 2.6]' we get L et. s. Now let us prove that Let. K. If possible let L <; K. We have the identity I t't Since R is simple, RxR = R for 0 i x E L. Therefore for any y E R, y = l.>,xr,. Then y' == L:rix's: = - L:r;xs;. Thus y - y' = L:(s;xr,+ r;xsi) E L. Siace y - y' covers K as y runs over R we get L == K. Hence theorem is established. Now let us generate a chain of left-o-ideals in R. Let U be a left-o-ideal in R. Then T(U) = {x E RIRox <; U} is also a left-o-ideal of R and then T 2 (U), T3(U), so. Therefore Ro U <; U <; T(U) <; T 2 (U) <; T 3 (U) <;. If U is maximalleft-o-ideal of R then either T(U) = R or U = T(U). Theorem 3.8 Let R be a prime ring. If S c T(S) or K c T(K) then R is commutative. Pmof. First of all let us prove a Lemma bellow. Lemma 3.9 If 1l is a prime ring and 0 i a E R satisfies the condition R 0 u = 0 then H.is colrunutative. Proof. If 0 a == 0 gives xa + a:r' = 0 If x E R. Now put x == xv, Y E R then we get xya + ay'x' = 0 which gives (xy + yx)a = O. Now again ---....[,.---- "
1~"Y. um.e,.., ~ ~... ' ~ TIte ~.Q/~