It is closed under addition and subtraction although it is sufficient to require closure under subtraction. Numbers of the form for and a fixed integer form a submodule since, for all ,. Given two integers and , the smallest module containing and is the module for their greatest common divisor ,. Beachy, J.
Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, Berrick, A. Birkhoff, G. A Survey of Modern Algebra, 3rd ed.
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Dates of meetings 12 September 10 October 14 November Wednesday , R92 Realfagbygget. Friday , R21 Realfagbygget. Examples of rings, algebras over a field, path algebra of a quiver, subrings, centre, characteristic, idempotents, nilpotent elements, direct product of rings, ideals.
Examples of ideals, ideals in matrix rings, intersection of ideals, ideal generated by a subset, principal ideal, principal ideal domain PID , quotient ring. A prime ideal is maximal in a PID, modules, examples, submodules, sums of submodules. Sums of submodules, submodules generated by subsets, linear independence, bases, free modules, module homomorphisms, external direct sum.
Quotient modules, fundamental theorem of R-homomorphisms, correspondence theorem, cyclic modules, Endomorphisms of a ring R regarded as an R-module.
Schur's lemma, Semisimple modules completely reducible modules , writing as direct sums, noetherian modules, noetherian rings. Characterization of noetherian modules; given a submodule, a module is noetherian if and only if a given submodule and its quotient are noetherian; Artinian modules. Artinian modules; artinian rings; characterization of artinian modules; a module is artinian if and only a given submodule and its quotient are artinian; a direct sum of artinian modules is artinian; a direct product of left artinian rings is left artinian; a nil left ideal in an artinian ring is nilpotent; endomorphism ring of a direct sum of modules.
Wedderburn-Artin theorem - proof of key lemma; Extended version of Wedderburn-Artin theorem. Non-elementary row and column operations; Smith normal form; Theorem: every matrix over a PID is equivalent to one in Smith normal form. Proof of theorem that every matrix over a PID is equivalent to one in Smith normal form; corollary that the row rank and column rank of a rectangular matrix over a ring coincide; discussion of Smith normal form in the case of Z and F[x]; examples. Structure theorem and proof for finitely generated modules over a PID; statement of uniqueness; application of structure theorem to the classification of finitely generated abelian groups; Introduction to rational canonical form.