## Prime Spectra in Non-Commutative Algebra |

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A E C'(o algebra Applications associated Azumaya algebra called closed coincide commutative completely consequence Consider construct contains contradiction Conversely COROLLARY crossed product defined definition denoted derive diagram division easily Edited element entails equal equivalent exact exists extends fact field final finite follows geometric given hence idempotent implies induced injective isomorphism K-algebra Ker f left ideal left Noetherian LEMMA localization maximal module Moreover morphism multiplicative Note o-torsion obtain Obviously OR-P pre-place prime ideal prime ring PROOF PROPOSITION prove pseudo-place Qo(R R-linear R-module reduction relation representing residue respect restricted ring homomorphism satisfies sheaf Spec statement structure Suppose symmetric kernel functor T-functor T-set Take Theorem theory tion topology torsion unique unramified pseudo-place valuation ring yields