Free Ideal Rings and Localization in General Rings
Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.
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admissible matrix atomic 2-ﬁr automorphism Bezout domain bound coefﬁcients Cohn columns comaximal comaximal relation coprime Corollary corresponding Deduce deﬁned deﬁnition denote eigenring embedding endomorphism equation equivalent ﬁeld of fractions ﬁltered ring ﬁnd finite ﬁnitely ﬁnitely presented ﬁr ﬁrst follows free algebra free associative algebras free module full matrix given hence holds homomorphism inﬁnite injective inner rank integral domain invariant element invertible isomorphism k-algebra lattice left factor left full left ideal left R-module Lemma local ring matrix ring monoid monomial multiplication n-ﬁr non-commutative non-full non-unit obtain principal ideal domain projective module Proof Proposition prove R-module result right ideal right invariant right Ore domain right R-module rows satisﬁes Section semiﬁr Show skew ﬁelds skew polynomial ring stably associated subalgebra submodule subring subset Sylvester domain Theorem torsion modules unique universal ﬁeld weak algorithm weakly ﬁnite zero
Page 531 - Since a module is projective if and only if it is a direct summand of a free module, we have: Corollary 1.6.5.
Page 542 - Bergman, Hereditary commutative rings and centres of hereditary rings, Proc. London Math. Soc. 23 (1971), 214-236. 4. J. Dauns and K.. Hofmann, The representation of biregular rings by sheaves, Math.
Page 521 - ... if we can pass from one to the other by a series of perspectivities.
Page 541 - Infinite primes and unique factorization in a principal right ideal domain. Trans. Amer. Math. Soc. 141 (1969), 245-254.
Page 522 - L is a distributive lattice with 0 and 1 in which every element has a complement, we can regard the process of associating with each element x its complement x' as a unary operator, ie an operator with one argument.