This is the only possible short answer i can think of, but it is not completely satisfying. Finitelygenerated modules over a principal ideal domain. Structure theorem for finitely generated modules over a principal ideal domain. An ideal generated by one element is called a principal ideal. Free modules the structure of modules 15 bibliography 17 index 19. If m is an r module, i is a 2sided ideal of r, and am 0 for all a 2i,m 2m, then we say that i annihilates m. Principal ideals can be counterexamples too if the ring isnt a domain. Tensor, tor, ucf, and kunneth colorado state university.
Our main result is the classi cation, up to algebraic and. There is a more general result in the same avor when mis an rmodule, and ris a principal ideal domain. Any finitely generated torsion d module has finite length. Then by the lemma, the ring is the intersection of its localization at height one primes. Awhere f is a free module and ais a projective ideal of r moreover, r is a principal ideal domain if and only if every. Question 1 suppose that g is a group of order pn, p a prime number, and that z is the center of g. Consequences of the structure theorem for nitely generated modules over a principal. This is a result on free ideal rings firs studied by p. In this handout we give prove a special case of the structure theorem for modules over a principal ideal domain pid, su cient for our needs at the outset in this course the general structure. All rings are commutative with identity unless speci. A characterization of sprime submodules of a free module. This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. In this paper we will characterize sprime submodules of a free module over a principal ideal domain. Similarly, for all n0, any subgroup of zn is a free abelian group.
Among these is your authors theorem that for a finite rank torsion free abelian group g there are, up to isomorphism, only finitely many groups isomorphic to summands of g see chapter 11. A ring r which is a principal ideal domain but such that rx is not a principal ideal do main. A module m over r is torsionfree if, for any r 2 r. Free modules over a principal ideal domain submodules of free modules are not always free. Let r be a principal ideal domain, let m be a free rmodule of finite rank n and. In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. It is reasonable to expect that the injective modules over a principal ideal domain also have a simple structure. A submodule ideal of r has the form rd, since r is a pid. Prime and maximal ideals mit opencourseware free online. An r module fis called a free r module if fis isomorphic to a direct sum.
A free module over an integral domain is torsionfree. Chapter 6 is devoted to the theory of modules over a principal ideal domain, establishing the cyclic decomposition theorem for finitely generated modules. In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain pid can be uniquely decomposed in much the same way that integers have a prime factorization. It is intended for undergraduate students taking an abstract algebra class at the juniorsenior level, as well as. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors e. Finitely generated modules over a principal ideal domain 6 5. Rmodule of rank n and n is a submodule of rank s, then there. In particular, every fvector space is a torsionfree fmodule. A free rmodule of rank s is the direct sum of s copies of r, each of which is noetherian as an rmodule when r is noetherian. Examples are aplenty in fact, if r is any pid that is not a. If i is a left ideal of a ring r, then i is a left rmodule with ra being the ordinary product in r. Therefore ba 6 0 for all b 6 0, and so a is a domain. Let r be a principal ideal domain and let m be an r submodule of r n.
Fundamental theorem of modules over a pid and applications travis schedler, womp 2007 september 11, 2007 0. Finitelygenerated modules over a domain in the sequel, the results will mostly require that rbe a domain, or, more stringently, a principal ideal domain. In particular, if ris an integral domain, then an ideal is free if and only if it is principal. We can now give some fairly wide classes of examples of noetherian and artinian rings. If you give some details about the proof, ill think good thoughts about you. In mathematics, a principal ideal domain, or pid, is an integral domain in which every ideal is principal, i. A is a submodule of a free module, and hence is free.
If p is a finitely generated projective ax, xxmodule and if axax. Modules over principal ideal domains let henceforth r denote a. In particular, if r is an integral domain, then an ideal is free if and only if it is principal. We say that m is a torsion module if annm 0 for all nonzero elements m m. Finitelygenerated modules over a principal ideal domain november 6, 2014 let rbe a commutative ring throughout. It doesnt make sense to talk about the ideal of an opensubset. Structure theorem for finitely generated modules over a principal.
Classification problems for shifts on modules over a principal ideal domain fabio fagnani and sandro zampieri abstract. Chapter 1 modules over a principal ideal domain in every result in. Any proper principal ideal in a finite commutative ring is a counterexample. Finitely generated modules over noetherian rings the goal of this section is to prove the following crucial theorem. Every principal ideal is a free module take x in r. The word domain means integral domain, and pid means principal ideal domain. Over a principal ideal domain a submodule of a free module is free. Fundamental theorem of modules over a pid and applications. For the rest of this section, r denotes a principal ideal domain. Amodule obtained from f via the operation of aon f through, that is for y2f. We will also investigate the properties of principal ideal domains and unique factorization domains.
There is a convenient way of picturing any submodule of a nite free module over a pid. The main aim of this note is to prove a baerkaplansky theorem for arbitrarily modules over principal ideal domains theorem 2. Further i is an additive subgroup of r, therefore ri is an abelian group with respect. Near to free modules are projective modules and flat modules cf. The twosided ideals of a principal ideal domain form a free commutative multiplicative semigroup with a zero and a unit element the maximal ideals of the ring are the free generators of this semigroup. Any nitely generated module over a noetherian ring is noetherian. Since ris commutative, there is no distinction between left, right and 2sided ideals. In constructive mathematics, many important rings may fail to be principal ideal domains in the naive sense. We call a domain a principal ideal domain pid iff every ideal is generated by one element. An nitely generated abelian group is isomorphic to.
Commutative algebra is the study of commutative rings and attendant structures, especially ideals and modules. A principal ideal a is free if it is generated by a non zero divisor. Uniform modules, primary modules, and notherlasker theorem. Every ideal of r is principal, hence nitely generated. For example z4z is free as z4z module, but 2z4z is a submodule that is not free. Prime and maximal ideals let r be a ring and let i be an ideal of r, where i r.
Qualifying examination in algebra spring 2018 answer each of the following seven questions clearly and concisely. For modules over a principal ideal domain, the analogous statement is the following. This follows just as in 7, theorem 4 once we observe that the following analogue of 7, theorem 3 is true. We might as well say that hamlet, prince of denmark is about a. Submodules of free modules over a pid the objective here is to prove that, over a principal ideal domain, every submodule of a free is also a free module and that the rank of a free submodule is always at least as large of the ranks of its submodules. Usually rwill be an integral domain and even a principal ideal domain, but these assumptions will be made explicitly. There is one situation where b is guaranteed to be a free amodule. An r module f is called a free rmodule if f is isomorphic to a direct.
Introduction the theory of modules is a rich subject that is very much a generalization of abelian groups. Z and q refer to the integers and the rational numbers respectively. This is not true for modules over an arbitrary principal ideal domain. As an example, both z and any eld are principal ideal domains, so the following. Finitelygenerated modules over a principal ideal domain emmanuel kowalski throughout this note, ais a principal ideal domain. Suppose that r is a dedekind domain with charr p 0 and. A nonzero ideal is a free module only when it is principal, since any two elements in an ideal are linearly related. If m is an r module and there exists an x in m such that m rx fax ja 2rg, then m is said to be a. Finitelygenerated modules over a domain in the sequel, the results will mostly require that r be a domain, or, more stringently, a principal ideal domain. Prove that if n is a normal subgroup of g, n 6f1g, then n \z 6f1g. First, for m r1 a free module on a single generator, an rsubmodule is an ideal in r. An ideal is isomorphic to the domain if and only if it is principal. There is a more general result in the same avor when mis an r module, and ris a principal ideal domain. Paper 2, section i 2g groups, rings and modules let r be an integral domain.
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