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. This is a result on free ideal rings firs studied by p. There is a more general result in the same avor when mis an rmodule, and ris a principal ideal domain. There is one situation where b is guaranteed to be a free amodule. This is not true for modules over an arbitrary principal ideal domain. Chapter 6 is devoted to the theory of modules over a principal ideal domain, establishing the cyclic decomposition theorem for finitely generated modules. There is a more general result in the same avor when mis an r module, and ris a principal ideal domain. All rings are commutative with identity unless speci. Prime and maximal ideals let r be a ring and let i be an ideal of r, where i r. Rmodule of rank n and n is a submodule of rank s, then there. Similarly, for all n0, any subgroup of zn is a free abelian group.
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. In particular, every fvector space is a torsionfree fmodule. This follows just as in 7, theorem 4 once we observe that the following analogue of 7, theorem 3 is true. If you give some details about the proof, ill think good thoughts about you. Finitelygenerated modules over a domain in the sequel, the results will mostly require that rbe a domain, or, more stringently, a principal ideal domain. Any nitely generated module over a noetherian ring is noetherian. An ideal is isomorphic to the domain if and only if it is principal. Chapter 1 modules over a principal ideal domain in every result in. Prove that if n is a normal subgroup of g, n 6f1g, then n \z 6f1g. Then by the lemma, the ring is the intersection of its localization at height one primes. Finitelygenerated modules over a principal ideal domain emmanuel kowalski throughout this note, ais a principal ideal domain. Amodule obtained from f via the operation of aon f through, that is for y2f. Classification problems for shifts on modules over a principal ideal domain fabio fagnani and sandro zampieri abstract. Finitelygenerated modules over a principal ideal domain november 6, 2014 let rbe a commutative ring throughout.
Fundamental theorem of modules over a pid and applications. An r module f is called a free rmodule if f is isomorphic to a direct. 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. Free modules the structure of modules 15 bibliography 17 index 19. Finitelygenerated modules over a principal ideal domain. The word domain means integral domain, and pid means principal ideal domain. First, for m r1 a free module on a single generator, an rsubmodule is an ideal in r. We call a domain a principal ideal domain pid iff every ideal is generated by one element. A ring r which is a principal ideal domain but such that rx is not a principal ideal do main. Commutative algebra is the study of commutative rings and attendant structures, especially ideals and modules.
Structure theorem for finitely generated modules over a principal ideal domain. 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. Our main result is the classi cation, up to algebraic and. 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. 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. A free module over an integral domain is torsionfree.
Any finitely generated torsion d module has finite length. Over a principal ideal domain a submodule of a free module is free. It doesnt make sense to talk about the ideal of an opensubset. A nonzero ideal is a free module only when it is principal, since any two elements in an ideal are linearly related. Fundamental theorem of modules over a pid and applications travis schedler, womp 2007 september 11, 2007 0. 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. It is possible for a free module to have a proper submodule of the same rank. Finitely generated modules over noetherian rings the goal of this section is to prove the following crucial theorem. Every finitely generated torsionfree module m over an integral domain a. Free modules over a principal ideal domain submodules of free modules are not always free. In constructive mathematics, many important rings may fail to be principal ideal domains in the naive sense. Therefore ba 6 0 for all b 6 0, and so a is a domain. Principal ideals can be counterexamples too if the ring isnt a domain.
A module m over r is torsionfree if, for any r 2 r. Structure theorem for finitely generated modules over a principal. In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. Structure theorem for finitely generated modules over a. This is the only possible short answer i can think of, but it is not completely satisfying. Question 1 suppose that g is a group of order pn, p a prime number, and that z is the center of g. Honors algebra 4, math 371 winter 2010 assignment 5 solutions for the problems 17, we. We say that m is a torsion module if annm 0 for all nonzero elements m m. A principal ideal domain pid is an integral domain in which every ideal is. In this paper we will characterize sprime submodules of a free module over a principal ideal domain. Tensor, tor, ucf, and kunneth colorado state university. This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. A is a submodule of a free module, and hence is free.
In particular, if ris an integral domain, then an ideal is free if and only if it is principal. On the other hand, a commutative ring is a principal ideal domain if and only if all of its nonzero ideals are free modules with unique rank. Uniform modules, primary modules, and notherlasker theorem. Qualifying examination in algebra spring 2018 answer each of the following seven questions clearly and concisely. Finitely generated modules over a principal ideal domain 6 5. In mathematics, a principal ideal domain, or pid, is an integral domain in which every ideal is principal, i. The submodule x is called an sprime submodule of m if for any ideal i of s and any fully invariant submodule u of m, iu. Modules over principal ideal domains let henceforth r denote a. Z and q refer to the integers and the rational numbers respectively. It is intended for undergraduate students taking an abstract algebra class at the juniorsenior level, as well as. A submodule ideal of r has the form rd, since r is a pid. Prime and maximal ideals mit opencourseware free online.
For example z4z is free as z4z module, but 2z4z is a submodule that is not free. Let r be a principal ideal domain, let m be a free rmodule of finite rank n and. A principal ideal a is free if it is generated by a non zero divisor. A characterization of sprime submodules of a free module. For modules over a principal ideal domain, the analogous statement is the following. As an example, both z and any eld are principal ideal domains, so the following. Let r be a principal ideal domain and let m be an r submodule of r n. An r module fis called a free r module if fis isomorphic to a direct sum. Usually rwill be an integral domain and even a principal ideal domain, but these assumptions will be made explicitly. An ideal generated by one element is called a principal ideal. However, the instructor may simply skim over this chapter, omitting all proofs. 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. Consequences of the structure theorem for nitely generated modules over a principal. Every principal ideal is a free module take x in r.
Since ris commutative, there is no distinction between left, right and 2sided ideals. If every submodule of every free amodule is free, prove that a is a principal ideal domain. We might as well say that hamlet, prince of denmark is about a. If p is a finitely generated projective ax, xxmodule and if axax.
More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors e. There is a convenient way of picturing any submodule of a nite free module over a pid. Any proper principal ideal in a finite commutative ring is a counterexample. We can now give some fairly wide classes of examples of noetherian and artinian rings. The main aim of this note is to prove a baerkaplansky theorem for arbitrarily modules over principal ideal domains theorem 2. Every ideal of r is principal, hence nitely generated. Paper 2, section i 2g groups, rings and modules let r be an integral domain. 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. Further i is an additive subgroup of r, therefore ri is an abelian group with respect. We will also investigate the properties of principal ideal domains and unique factorization domains. 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.
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