PRIMALITY TEST IN ITERATED ORE EXTENSIONS. JOS ´ E L. BUESO, J. G ´ OMEZ-TORRECILLAS, F. J. LOBILLO, F. J. CASTRO In noncommutative Algebraic Geometry the points of the most useful spectrum are the two–sided prime ideals. An interesting computational problem is to decide if a given ideal is prime or not. In commutative polynomial rings there is the Gianni et al. test [2]. In this paper we generalize this test for completely prime ideals in a wide class of iterated Ore extensions of a field. There are interesting examples of algebras where all prime ideals are completely prime. The classical example, the universal enveloping algebra of a finite dimensional solvable Lie algebra over a field of characteristic zero, is stated by Dixmier–Gabriel theorem. Sigurdsson extended this result to an iterated differential Ore extension of a commutative noetherian –algebra. Recently, [4] have improved previous assertion to more general iterated Ore extensions with non–trivial automorphism, including the quantum examples O q ( n ), O q (M n ( )), O q (GL n ( )) and O q (SL n ( )) for q not a root of unity (also U + q ( ) has been solved by [7]). These are examples of solvable polynomial rings. From the viewpoint of Noncommutative Algebra, an essential property of these algebras is the Poincar´ e–Birkhoff– Witt theorem. Thus, our approach to the notion of solvable polynomial ring, introduced in [5], emphasizes this property. In section 1, we recall the algorithmic treatement of Gr¨ obner bases in this context. The primality test is described in section 2. Our test involves some computations in the classical quotient ring of a noncommutative domain. So, section 3 is devoted to solve some computational problems arising in this noncommutative setting. In section 4 we describe a general procedure deciding if a given ideal is completely prime. In section 5 the previous procedure is in fact a primality test for ideals in iterated differential operator rings. In section 6 the procedure of section 4 produces a test of complete primality (resp. a primality test) for ideals in the coordinate ring of quantum spaces (resp. if the entries of the matrix of parameters generate a torsionfree group). References [1] J. L. Bueso, F. J. Castro, J. G´omez Torrecillas, and F. J. Lobillo, An introduction to effective calculus in quantum groups, Rings, Hopf algebras and Brauer groups. (S. Caenepeel and A. Verschoren, eds.), Marcel Dekker, 1998, pp. 55–83. [2] P. Gianni, B. Trager, and G. Zacharias, Gr¨ obner bases and primary decomposition of polynomial ideals, J. Symb. Comput. 6 (1988), 149–167. [3] K. R. Goodearl, Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), 324–377. [4] K. R. Goodearl and E. S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994), 1017–1025.