## Commutativity of rings |

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a p(a absolutely algebraic algebraic closure algebraic extension algebraic over F an^a characteristic p^0 contains contradiction Corollary Dedekind domain Definition Let division algebra division ring element a e element p(a exists an element exists an integer finite field finite number finitely many elements Hence idempotent implies infinite integer n>0 integral domain integrally closed irreducible polynomial isomorphic K-ring Lemma Let F Let F(x m-system minimal ideal minimum polynomial monic natural number nil ideal nilpotent elements non-trivial valuations non-unit non-zero radical number of prime pick prime field primitive rings proper prime ideals Proposition quotient field R/N is commutative ring with center satisfies separable over F Step subdirect sum subdirectly irreducible ring Suppose transcendence basis transcendental over F two-sided ideal unramified yc-w z(ab zero divisor zero radical