(A) INTRODUCTION Let R denote a commutative ring with identity and (R) denote the ring of n* n matrices over R . For 90 years much effor t has been devoted to the following question. Suppose P(A) is an invariant defined on matrices A in (R) . Then, determine the set of R-linear mappings T : (R) -*• (R) which preserve the invariant, i . e . , find the linear mappings T satisfying p(T(A)) = P(A) for all A in (R) . For example, the invariant might be the determinant, i . e . , p(A) = det(A) , and here we would seek all linear mappings T with det(T(A)] = det(A) . The case of the determinant was f i r s t studied where R = £ , the complex numbers, by Frobenius in 1897 [1]. Frobenius showed that i f T : ((£) •+ (CC) preserved determinants, then either T(A) = PAQ for all A in (C) or that T(A) = PA Q for al l A in (C) where P and Q are invertible matrices with det(PQ) = 1 . Many of the questions regarding linear maps T : (R)n - (R) which preserve invariants can be reduced to the problem of determining the set of linear maps which carry the rank one matrices into themselves. This fact was noted by M. Marcus in a survey articl e [10] in 1971 on this subject where R was assumed to be a f i e l d . In 1959, assuming that k is an algebraically closed fiel d of characteristic zero, then Marcus and Moyls [12] proved that, i f T : ( k ) " v ( k ) n is a k-linear mapping with the property that rank(T(A)) = 1 whenever rank(A) = 1 , then T has the form T(A) = PAQ for all A in (k) or T(A) = PA^ for all A in (k) where P and Q are invertible matrices. I f one examines carefully the proofs of Marcus and Moyls [12], one finds that their proof carries over to. any hloJLd o& any ckaKao£vva- tic. The Marcus and Moyls result was proven by the use of multilinear algebra. An elementary matrix theoretic proof of the same result was given by Mine [17] Received by Editor Dec. 7, 1981 1
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