Quotient Rings and f-Radical Extensions of Rings

Abstract

By testing quotient rings, we give another viewpoint concerning the relationship between PI and Goldie properties, etc., and f-radical extensions of rings. The main result proved here is as follows: Let R be a prime algebra without nonzero nil right ideals. Suppose that R is f-radical over a subalgebra A, where f(X ₁,…, X _t ) is a multilinear polynomial, not an identity for p × p matrices in case char R = p > 0. Suppose that f is not power-central valued in R. Then the maximal ring of right (left) quotients of A coincides with that of R. Moreover, R is right Goldie if and only if A is.

Key Words:

• f-Radical extension
• GPI
• PI
• Prime algebra
• Quotient ring

2000 Mathematics Subject Classification:

• 16A38
• 16N60
• 16R50
• 16U80

Notes

Communicated by E. R. Puczylowski.

^♯Members of Mathematics Division, National Center for Theoretical Sciences at Taipei.