Polynomial Rings with a Pivotal Monomial1
نویسنده
چکیده
1. Amitsur in his paper on Finite Dimensional Central Division Algebras [l] has proved that in a division ring D with center C, (P: C) 5= ra2 < =o if and only if every primitive homomorphic image of a polynomial ring P[x] is a complete matrix ring Ah, h^n, over a division ring A. Equivalently speaking, a division ring is finite dimensional over its center if and only if the polynomial ring over it has a J-pivotal monomial (written as JPM). The object of this note is to show that if R is a ring with a nilpotent (Jacobson) radical then the polynomial ring P[x] has a JPM if and only if R[x] has a polynomial identity. Amitsur's result then follows as a special case of our result. Our proof of Theorem 1, in obtaining sufficiency, is on the same lines as that of Amitsur. 2. We begin with
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