Free Rings and Their Relations

a₁ a₂ ACC₁ atomic 2-fir atomic factors automorphism b₁ Bergman 67 Bezout domain Clearly coefficients Cohn column comaximal comaximal relation COROLLARY corresponding cyclic defined degree denote determinantal sum direct sum direct summand division algorithm eigenring embedding endomorphism equation equivalent exists field of fractions filtered ring finitely follows free algebra free associative algebra free module given hence homomorphism indecomposable integral domain invariant elements inverse weak algorithm isomorphism left factor Lemma local ring mapping Math maximal monomials multiplication n-fir n-generator non-unit non-zero element obtain prime matrix ideals principal ideal domain principal right ideal projective Proof Proposition prove quotient R-field R/aR R/CR R₁ rank result right ACC right linearly right Ore ring right R-module right v-dependent satisfies semifir semigroup Show skew polynomial ring subalgebra submodule subring Theorem torsion module u₁ unique x₁ zero