Infinite Dual Goldie Dimensions Of Modules

Dimensions play an important role in the study of rings and modules. For example,the number of vectors contained in the base of a vector space V over a filed K is defined asthe dimension of V, denoted by dim（V）, which is an important arithmetic index of vectorspace. We also have known that if a module M is both Notherian and Artinian, then itslength, denoted by c（M）, can be regarded as its dimension. In particular, if M is a vectorspace with finite dimension, then its length equals to the dimension of the vector space.Moreover, we can define injective dimensions and projective dimensions of a module Mby using injective resolutions and projective resolutions of modules in homological algebra,respectively. These two kinds of dimensions can be used to show the diference between amodule M and injective modules and projective modules.Grezeszczuk and Puczy owski defined finite dual Goldie dimensions of modules onthe basis of previous studies by Fleury and other scholars. This dimension is defined to beinfinite if it is not finite. But We have known that there are many kinds of “infinity” in settheory such as countable infinity and continuum, so the use of this tool was limited. It issignificant to generalize dual Goldie dimensions of modules to infinity.First, we define dual Goldie dimensions of modules:Definition2.1Let M be a module. The dual Goldie dimension of a module M, denotedby codim（M）, is defined as the supremum of the length of coindependent submodules in M.That iscodim（M）=sup{|I|;（Ni）_i∈Iis coindependent in M}.It is easy to see that in finite circumstance Definition2.1is coincident with dual Goldiedimensions defined in related literatures. Thus, it can be regarded as a generalization of finitedual Goldie dimension.Infinite dual Goldie dimensions satisfy the following three properties which generalizethe corresponding results of finite dual Goldie dimensions. Proposition2.1For a module M, if M=M₁⊕···⊕M_k, thencodim（M）=codim（M₁）+···+codim（M_k）.Proposition2.2Let N be a submodule of M. If N M, then codim（M）=codim（M/N）;conversely, if codim（M） is finite and equals to codim（M/N）, then N M.Proposition2.3Let N be a submodule of M. If N is a weak supplement of L in M, thencodim（M）=codim（M/L）+codim（M/N）.Then, we show what a cardinal number is attained in a module M means. That isDefinition3.1Let M be a module. Given a cardinal number, we say is attained inM if M contains a family of coindependent submodules （Mα）α∈Iwith|I|=.We also proveTheorem3.1For a ring R, if codim（R）=0when R is regarded as a left regular module,then0is attained in R.Theorem3.2Let M be a finitely generated module. If codim（M）=0, then codim（M）is attained in M.At last, we give some examples which show that for some special modules, their dualGoldie dimensions can be attained in themselves.Example3.1LetZbe the ring of integers. Then codim（Z）=0whenZis regarded asa left regular module and codim（Z） is attained inZ.Example3.2LetQ[X] be the ring of polynomials over the field of rational number.Then codim（Q[X]）=0whenQ[X] is regarded as a left regular module and codim（Q[X]） isattained inQ[X].Example3.3Let M=α∈IMα, I is infinite and codim（Mα）<∞for everyα∈I, thencodim（M）=|I|and codim（M） is attained in M. In particular, it is also true for every Mαishollow or simple.