On P.I. ring and standard identities

An example is given of a ringR (with 1) satisfying the standard identityS ₆x ₁, ...,x ₆] butM ₂(R), the 2 × 2 matrix ring overR, does not satisfyS ₁₂x ₁, ...,x ₁₂]. This is in contrast to the caseR=M _n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS _2n x ₁, ...,x _2n] andM ₂(R) satisfiesS _4n x ₁, ...,x _n]. Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.