Kaplanskys theorem for ufds new mexico state university. Mcgovern, gena puninski, and philipp rothmaler abstract. It also shows that a summand of a direct sum of cyclic modules over a commutative local ring is again a direct sum of cyclic modules. Thus, it is natural to ask if kaplansky s theorem on projective modules has a w module theoretic analogue. In abstract algebra, kaplansky s theorem on projective modules, first proven by irving kaplansky, states that a projective module over a local ring is free. Projective modules over the endomorphism ring of a biuniform module gena puninski abstract.
We would like a means to recognise projective modules p without having to consider all possible surjections and morphisms from p. We apply this result to obtain a number of results on direct sums of modules, including generalizations of kaplansky s theorem 4 that a projective module over a logical ring is free. Projective modules, idempotent ideals and intersection. Projective modules were first introduced in 1956 in the influential book homological algebra by henri cartan and samuel eilenberg. Projective modules over dedekind domains, february 14, 2010 3 2. Recall that a module a4 of a4 is called pure projective if it is a direct summand of a direct sum of finitelypresented modules. Besides the class of projective modules, the wellknown kaplansky classes are the class of injective and also the class of at r modules. Such a ring r could also be characterized as any finitely generated flat module is projective. Kaplansky proved the general case by proving it for primary modules over commutative complete discrete valuation rings. It is easy to argue that the class of injective modules and the class of flat modules are kaplansky see 9, lemma 5. Generalizing the baerkaplansky theorem sciencedirect. Actually, the notion of w projective modules appeared. We begin by noting that theorem 3 is known when p is finitely generated 3, ch. Projective diagrams over partially ordered sets are free.
By a theorem of kaplansky every projective module is a direct sum of countably generated ones. The goal of this section is to prove that a module is projective if and only if it is flat, mittagleffler, and a direct sum of countably generated modules theorem 10. Vasconcelos1 one of the aims of this paper is to answer the following question. Actually, the notion of wprojective modules appeared. Projective modules over universal enveloping algebras september 27, 2011. A class kof r modules is called a kaplansky class if there exists a cardinal number such that for every. We prove the cancellation theorems for projective modules and also prove the qualitative version of laurent polynomial analogue of horrocks theorem. Kaplanskys theorem for ufds mathematics 581, fall 2012 in this note we prove a theorem of kaplansky which gives a necessary and su cient condition for an integral domain to be a ufd. Noncommutative rings michael artin class notes, math 251, berkeley, fall 1999. In this paper, we give a characterization of left kcyclic rings.
Of course, any projective module is pure projective. I g i, where each g i is a reduced group in the class g of selfsmall mixed group of finite torsionfree rank. A module is said to be projective if it is a direct summand of a free module. We characterize rings over which every projective module is a direct sum of. We investigate the relationship between projective modules and idempotent ideals for group rings, polynomial rings and more general rings, giving a survey of known results, proving some new results and raising a number of questions. The main aim of this note is to prove a baer kaplansky theorem for arbitrarily modules over principal ideal domains theorem 2. Projective modules over universal enveloping algebras september 27, 2011 3 14. Is the kaplansky theorem equivalent with the axiom of choice or it can be proved with the weaker axiom i. Thus, it is natural to ask if kaplanskys theorem on projective modules has a wmodule theoretic analogue. Recently i got the chance to read and understand i. Kaplansky made major contributions to group theory, ring theory, the theory of operator algebras and field theory and created the kaplansky density theorem, kaplansky s game and kaplansky conjecture.
Kaplansky a module is projective if and only if it is c. Projective modules, idempotent ideals and intersection theorems. This leads to several complete cotorsion theories in such. Projective modules over dedekind domains, february 14, 2010.
Projective modules with finitely many generators are studied in algebraic theory. Pdf a wellknown theorem of kaplansky states that any projective module is a direct sum of countably generated modules. Assuming the axiom of choice, then by the basis theorem every module over a field is a free module and hence in particular every module over a field is a projective module by prop. Introduction a wellknown theorem of kaplansky states that every projective module over a local ring is free. Commutative algebra is the study of commutative rings. This is the only possible short answer i can think of, but it is not completely satisfying. If r is left hereditary, then every submodule of a free left r module. One of the most basic results in commutative algebra, given as the. Thus, a ring over which every finitely generated right module is free is, in a sense, ever so close to having all projective right modules free. We give a procedure and describe an algorithm to compute the dimension of a module over laurent polynomial ring.
Separative cancellation for projective modules over. In stark contrast, almost projective modules modules possessing a rich supply of small projective submodules have a very complex structure in general. Commutative algebra is the study of commutative rings and attendant structures. By reformulating the baer kaplansky theorem it is shown that it holds for large classes of modules over arbitrary rings. Kaplanskys theorem on projective modules wikipedia. In this section we introduce a devissage technique for decomposing a module into a direct sum. Projective modules over universal enveloping algebras. Doing so will lead us to the notion of strongly kaplansky classes of modules, which we will need in order to prove theorems 3.
Nakayamas lemma and krulls intersection theorem all rings are unital, commutative rings. Our main theorem, theorem a, is an extension to the nonprojective case of a strong version of serres famous theorem on free summands of projective modules. We might as well say that hamlet, prince of denmark is about a. In this paper, we prove the wversion of this theorem, where w is a hereditary torsion theory for modules over a. It follows from a combination of results of fuchs, kaplansky and handelman that the regular rings whose finitely generated projective modules satisfy c are precisely those with stable rank one cf. The class consists of arbitrary direct sums of the form g. Foundations of module and ring theory a handbook for study and research robert wisbauer university of dus. By a theorem of kaplansky every projective module is a direct sum of. Another result of kaplansky says that, over any ring, any projective right module is a direct sum of countably generated projective modules. We investigate in nitely generated projective modules over the endomorphism ring of a biuniform module. Noncommutative rings michael artin mit mathematics. On projective modules and computation of dimension of a. Suppose first that is projective and let by corollary 1, there exists an module and a free module such that but then and thus is projective.
A projective resolution always exists since one can easily construct a resolution by free modules. In particular, we determine several classes of rings over which every pure projective left module is a direct sum of cyclic modules. Then is projective if and only if is projective for all. We study modules which are direct sums of indecomposables. A natural source of local rings is the class of endomorphism rings of indecomposable modules. A ring on which all finitely generated projectives modules. New classes of modules satisfying the classical baer kaplansky theorem are. We present evidence which strongly suggests that if a ring has noncentral idempotents then no reasonably diverse class of its modules will satisfy the original baer kaplansky theorem. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a polynomial ring this is the quillensuslin theorem. Kaplansky k proved that there exists a countably generated summand of p that contains x. This result was recently extended to exchange rings by yu 50, theorem 9. If they know maschkes theorem, one can be a bit more refined and show that in the modular case when the trivial module has infinite projective dimension, so there one has a wide example of nonhereditary algebras which one. In this article we provide arguments for constructing kaplansky classes in the category of complexes out of a kaplansky class of modules. Let a be a commutative ring for which projective ideals are finitely generated.
Projective modules over local rings before we can consider projective modules over dedekind domains, we will consider the case of projective modules over noetherian local rings. We also prove rank versions of these results for torsionfree modules over commutative domains. We consider baerkaplansky type theorems for the class. We show that there is a generalized baer kaplansky theorem for all such modules with some additional conditions. Oct 28, 2011 examples of projective modules posted. Nakayamas lemma the following is one of the most basic results in commutative algebra, known as. Example of a projective module which is not a direct sum. A projective resolution of m is a complex p of projective modules such that hip 0 for i0 and h0 p. Kaplansky was the doctoral supervisor of 55 students including notable mathematicians hyman bass. The main aim of this note is to prove a baerkaplansky theorem for arbitrarily modules over principal ideal domains theorem 2. The representation theory of nite groups has a long history, going back to the 19th century and earlier. It has as its immediate consequences in the projective case basss. In this paper, we prove the wversion of this theorem, where w is a hereditary to.
Journal of pure and applied algebra 20 1981 712 northholland publishing company projective diagrams over partially ordered sets are free michael hpner and helmut lenzing fachbereich mathematik, unioersitgesamthochschule paderborn, d4790 paderborn, germany a generalization of kaplansky s theorem on projective modules over local rings is used to show that every projective. Let r be a commutative semihereditary ring, p a projective r module. Third problem set due monday, april 18 a ring is left hereditary if every left ideal is a projective module. Commutative algebra is the study of commutative rings and attendant structures, especially ideals and modules.
By a classic theorem of kaplansky, the structure theory of projective modules over an arbitrary ring reduces to the countably generated ones. The following lemma provides this, and shows that the. The next results will be used to prove the existence of preenvelopes relative to kaplansky classes. He published more than 150 articles and over 20 mathematical books. Projective modules over dedekind domains, february. Then addm will denote the full subcategory of the category of right r modules whose objects are direct.
We also show a lazardlike theorem for flat quasicoherent sheaves for quasicompact and semiseparated schemes which. We prove a version of kaplansky theorem for quasicoherent sheaves, by using drinfelds notion of almost projective module and the hill lemma. The main result is that a projective module is a direct sum of countably generated modules theorem 10. Kaplansky s theorem, asserting that every projective module is a direct sum of projective modules with countably many generators, reduces the study of the structure of projective modules to the countable case. A wellknown theorem of kaplansky states that any projective module is a direct sum of countably generated modules. Projective diagrams over partially ordered sets are free michael hdppner and helmut lenzing fachbereich marhemarik, llniversiriirgesam. Research article chunhua yang kaplansky classes of.
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