Generahzed Identities and Semiprime Rings with Involution 57

then M,M*,fGM. So Rj^ is a prime ring with normal-in-zero nonpositive involution * (see Lemmas 2.2.2 and 2.2.5). By Montgomery's theorem 2.1.1, Rj^ satisfies S^(X) = 0. Let cp^.R-^Rj^ be the canonical homomorphism. Then (рм{^Щ satisfies S^{X) = Q. If weM then wRç^MR. So (p^{wR) = Q (see [6], Lemma 1). Thus in each case (Рм{^Щ satisfies S^{X) = f). But the set of momorphisms (p^, MeM{C), defines an embedding

wR - ^ П (>^^)m

MeM ( C )

( see [5]). Therefore wR (and so wA) satisfies S^{X) = 0. Lemma 2.2.10. With the notation of 2.2.9

wR = Q{wÄ).

Proof Let I = I{w). Then wI^A, (\-w)I^A and / = w/ + (l-w)/ is a dense ideal of A and is a direct sum of ideals wl and (1 — w)/. Therefore

o = ö(^) = 0(/) = 0(w/)eß((l-w)/) = wße(l-w)ß.

Similarly ,

R { A ) = wR®{l-w)R = R{wI)®R{(l-w)I).

Since w/çwR is a semiprime Pl-ring (see Lemma 2.2.9), by [33], Theorem 6,

R { wI ) = Q{wI). Since w/çwtÎçwJR,

Q { wÄ ) = wR.

2 . 2 . 11 . Since Ä = wRe(l-w)Ä and C=C{R) then C(wi^) = wC (see ma 1.3.1), If MgM(wC) then

M' = M-^{i-w)CeM{C) and {wR)m=Rj^,.

But 1—w = w-I-m* + i?gM and the idempotents u,u'^,v are orthogonal, so u,ti*, i;gM and Rj^ is a prime ring with nonpositive normal-in-zero involution *. By [29] or [19] the complete ring of quotients of a semiprime Pl-ring is regular and so wR is a regular Pl-ring. By [6], Lemma 1,

{ wR ) m = {wR)IM{wR).

Since a prime homomorphic image of a semiprime regular Pl-ring is a simple ring, {wR)^ is a simple ring (see, for example, [20], Theorem 2.3). By the above remarks and by Montgomery's Theorem 2.1.1, {wR)j^ is the (2x2)-matrix ring over a field with symplectic normal involution. Let

A^=uÄ , A2 = u*Ä, Ay = vÄ, A4, = wÄ, B = wR, K=C{B) = wC.

Note that Л^, Л2,^3 and A4, have the properties a) and b) of Theorem 2.2.1.