Hopf crossed products, rings of quotients, and prime ideals
This interesting article is concerned with crossed products \(A = R \#_\sigma H\) of a Hopf algebra \(H\) over an algebra \(R\). The authors introduce two new ``actions'' of \(H\) on \(R\) which are extremely useful in simplifying computations involving the cocycle \(\sigma\). Using this technique, the original \(H\)-action on \(R\) is extended to various quotient rings of \(R\) and corresponding quotient rings of \(A\) are described. These results are then applied to study prime ideals of \(A\). The main result in this direction is the following (Theorem 7.12): Let \(H\) be semisimple and either commutative or cocommutative and assume that \(R\) is \(H\)-prime. Then \(A\) is semiprime, with at most \(\dim H\) many minimal primes, and a prime of \(A\) is minimal precisely if it intersects \(R\) trivially. Replacing the above assumption on \(H\) by the hypothesis that the \(H\)- action on \(R\) is inner, in a suitable sense, an analogous but slightly weaker result is obtained (\(A\) need not be semiprime). Various sufficient conditions for \(A\) to be semiprime are derived which greatly extend what was previously known on this subject, but the general question, as to whether \(A\) is semiprime if \(R\) is \(H\)-semiprime and \(H\) is semisimple, remains open.
scientific article; zbMATH DE number 755386