THE EQUATION x'mXd-dx**b

for all #, y in 31 and all a in $. The constants relative to such a derivation are the elements of 3t that commute with d. We shall call an element b a d-integral if b~a for some element a in 31, that is, if the equation x' ~xd—dx*=b has a solution in 31. Clearly if a is a solution of this equation then the totality of solutions is the set {a+c} where c ranges over the set of d-constants. In a recent paper appearing in this Bulletin, R. E. Johnson obtained a necessary and sufficient condition that an element b be a d-integral under the assumption that 31 is a separable algebraic division ring. In this note we allow 3Ï to be an arbitrary algebra but we make the assumption that d is an algebraic element in the sense that it satisfies a polynomial equation with coefficients in 3>. We obtain a necessary condition, which is equivalent to Johnson's condition when 31 is a division ring, that b be a d-integral. If the minimum polynomial j*(X) of d is relatively prime to its derivative M'(X), then it is easy to see that the condition is also sufficient and one may give an explicit formula for a solution of the equation x' ==&. If we assume that 31 is a simple algebra satisfying the descending chain condition for left ideals then we can show that our condition is also sufficient when JW(X) is a product of distinct irreducible factors in $[X] and in certain other cases. Here, however, we do not display a solution but merely prove its existence. Our results include, of course, Johnson's result for algebraic division rings, since the minimum polynomial of an element in such a ring is irreducible. No assumption about separability is required. In order to obtain a condition for the solvability of the equation x~b we consider the matrices