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LetRbe a prime ring andδaσ-derivation ofR, whereσis an automorphism ofR. Our aim (Theorem 1) is to determine any automorphismSofR[t;σ,δ] satisfyingS(A)⊆Rfor a one-sided idealA≠0 ofRby dropping the assumptiont∈R[t;σ,δ]. As an application, ifRis a domain, it is shown that any X-inner automorphism ofR[t;σ,δ] stabilizesR. [ABSTRACT FROM PUBLISHER]

LetRbe a right Ore domain and φ a derivation or an automorphism ofR. We determine the right Martindale quotient ring of the Ore extensionR[t; φ] (Theorem 1.1). As an attempt to generalize both the Weyl algebra and the quantum plane, we apply this to ringsRsuch thatk[x] ⊆ R ⊆ k(x), wherekis a field andxis a commuting variable. The Martindale Quotient quotient ring ofR[t; φ] and its automorphisms are computed. In this way, we obtain a family of non-isomorphic infinite dimensional simple domains with all their automorphisms explicitly described. [ABSTRACT FROM AUTHOR]

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(X1,..., Xt) 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. [ABSTRACT FROM AUTHOR]