Modules Whose Endomorphism Rings Are Division Rings

Indecomposable modules are very important in representation theory of algebras and the theory of rings and modules.As a specific case,modules whose endomorphism rings are di-vision rings have attracted wide attention.Note that the endomorphism ring of any indecomposable A-module is a field for some special algebra A over an algebraically closed field K.However,when the base field K is not algebraically closed,the endomorphism ring of an indecomposable A-module need not be a division ring and,in general,it is difficult to decide all indecomposable A-modules.Therefore,we turn to consider the modules whose endomorphism rings are division rings over some path algebras.On the other hand,in the theory of rings and modules,it is known as the Schur's Lemma that the endomorphism ring of any simple module is a division ring.But as we all know,the converse of Schur's Lemma does not hold in general.Just as the case that simple modules are frequently adopted to characterize certain rings,those modules having division endomorphism rings can also be used in characterizing rings such as rudimentary rings,which can be regarded as a generalization of primitive rings.We will carry out further investigation on rudimentary rings in this thesis.Our research include the following two aspects.(I)Modules with division endomorphism rings over some path algebras.Since the category of modules over a path algebra KQ is equivalent to that of K-linear representations of the quiver Q(where K is a field or a commutative ring),and the later is more convenient to deal with,so we often adopt the representations instead of the modules as usual.Firstly,let K be a field;Q a finite,connected,and acyclic quiver with at least two points;?s the reflection at a sink a in Q;Q' = ?aQ;S_a~+:repK(Q)? repK(Q')the corresponding reflection functors.By virtue of a theorem of I.N.Bernstein,I.M.Gel'fand and V.A.Ponomarev,if M is an indecomposable representation of Q such that S_a~+(M)? 0,then End(M)? End(5a+(M)).We provide a direct proof for this result.Secondly,given a Kronecker quiver Q and a commutative ring A,if M is a module over the path ring AQ such that EndAQ(M)is a division ring,it is proved that M becomes a module over the path algebra FQ and EndAQ(M)? EndFQ(M),where F = Q(A/P)is the field of fractions of A/P and P is the prime ideal of A induced by M.Then,for an arbitrary field K,we give a sufficient and necessary condition under which the endomorphism ring of the representation K2(?)K2 is a division ring.Thirdly,for the quiverQ(?)with one point and one loop,all representations of Q(over an algebraically closed field K)with division endomorphism rings are obtained based on some known results.For the quiver Q:1(?)2 and an arbitrary field K,we figure out when the endomorphism ring of a representation K2(?)Km(m ? 1)is a division ring.Finally,the endomorphism rings of some representations for two wild quivers are considered.We give some examples of representations whose endomorphism rings are division rings.(II)Further study on rudimentary rings.As a matter of fact,the notion of rudimentary rings is a generalization of primitive rings.Inspired by the results on primitive rings and primitive ideals,we introduce rudimentary ideals and prove that an ideal I in a ring R is right rudimentary if and only if I is the annihilator of a right R-module whose endomorphism ring is a division ring.We also characterize right rudimentary ideals of full matrix rings over R.In addition,we consider the condition for an ideal I of a ring R to be a right rudimentary ring(which might have no identity).Let R be a ring and I an ideal which contains a nonzero central element of R.If V is a faithful right R-module such that EndR(V)is a division ring,we prove that EndR(V)= EndI(V).Then the relation between the class of rudimentary rings and that of prime rings is investigated.We give a sufficient condition for a right rudimentary ring to be prime.Finally,*-rudimentary rings are defined and it is shown that right*-rudimentary property is Morita invariant.