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The basic properties of associative rings R satisfying a polynomial identity p[x 1..., xn ] = 0 were obtained under the assumptions that the ring was an algebra [e.g.,  Ch. X], or with rather strong restrictions on the ring of operators (). But it is desirable to have these properties for arbitrary rings, and the present paper is the first of an attempt in this direction. The problem is almost trivial for prime or semi-prime rings but quite difficult in arbitrary rings. The known proofsdoi:10.1017/s0027763000011909 fatcat:tqg24xrogrbdvlihtbzjuv27mm