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PROOF. Since Q?=ID p (where I denotes the identity operator and e denotes the linear span of I with p1) it suffices to prove only the second statement. Let T E p1 Then B(T) E qp and so if F is a scalar homomorphism on p F(B(T))=O by Theorem 3.2. Since (V(T))2=limi Xi(T)V(T), either F(V(T))=O or F(V(T))=limi Xi(T). Suppose, if possible, that F(V(T))= limi Xi(T) (=X(T))$O. Choosej so that X3(T)$O. Define Sin p1 by Sj= 1 and Si, 3+1 =-X(T)IXj(T) (i= 1, 2, 3, * * ). Then SV(T) is the zero operator and so F(S)=O (because F(V(T))$0). On the other hand, S= V(S)+B(S) with B(S) E p and V(S)=E. Thus F(B(S))=O and F(V(S))$O because V(T)E= V(T) and F(V(T))$O by assumption. Hence F(S) cannot be zero. This contradiction proves that F(V(T)) must be zero and hence that F is identically zero.

Corrections to: “The structure of 𝐵[𝑐] and extensions of the concept of conull matrix”