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Published online by Cambridge University Press: 17 June 2002
Let R be a prime ring with extended centroid C, $\rho$ a non-zero right ideal of R and let $f(X_1,\dots,X_t)$ be a polynomial, having no constant term, over C. Suppose that $f(X_1,\dots,X_t)$ is not central-valued on RC. We denote by $f(\rho)$ the additive subgroup of RC> generated by all elements $f(x_1,\dots,x_t)$ for $x_i\in\rho$. The main goals of this note are to prove two results concerning the extension properties of finiteness conditions as follows.
(I) If $f(\rho)$ spans a non-zero finite-dimensional $C$-subspace of $RC$, then $\dim_CRC$ is finite.
(II) If $f(\rho)\ne0$ and is a finite set, then $R$ itself is a finite ring.
AMS 2000 Mathematics subject classification: Primary 16N60; 16R50
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