Modules Graded By Generalized G-sets And Smash Products

C.Nastasescu and F.Van Oystaeyen introduced rings andmodules graded by groups in their book (Graded Ring Theory)); the two authors generalized the concept of modules graded by groups to that of modules graded by G-sets in the paper (Modules Graded by G-sets)), and gained a series of results on that basis.We study generalized transformation groups in this paper, and use the concept of generalized transformation group to generalize the concept of group action to that of generalized group action. Based on this, we generalize modules graded by G-sets to modules graded by generalized G-sets.We also study some properties on the smash products of graded rings (modules graded by G-sets) and G-sets in this paper.Generalized transformation group is the generalizing of the concept of transformation group. We discuss the conditions under which a set of common transformations (may be non-bijective) becomes a group, the main results are:Theorem 1.1.6 For a partition J of a setS, we choose T as s set of representatives of I, then the set of induced transformations of all the hijective transformations from J to I constructs a group, the composition is the compositionof transformations, denoted by G,.To distinguish with the concept of transformation group, we call G, and the subgroups of G generalized transformation groups.Theorem 1.1.7 If a set Mof transformations of anon-emptyVset S constructs a group, and the composition is the composition of transformations, then N is a generalized transformation group.Theorem 1. 1.8 Let G he a generalized transformation group.If C has a hijcctive transformation, then G only has hijective(1transformations; If C has an-bijective transformation,then G has non-hi jective transformations only.An action of a group C on a set A is equivalent to a group homomorphism from C to a group of transformations of A. In chapter 1 section 2, we use the concept of generalized transformation groups to generalize the concept of group action to that of generalized group action, then we have the concept of generalized C-sets. The main result of this part is:Theorem 1.2. 1 A generalized group action of a group C on a ^et S determines a group homomorphism fromG to a general i7^ed transformation group of S uniquely. Conversely, a homomorphism from G to a generalized transformation group determines a unique generalized group action of G on S.Assume R=$^EG R is a graded ring, M=ece(;MJ is agraded module over R, then for Vr^, E R^,. , rn Eu,rEG, satisfying r^m E . Regard the product as G act on G, from this notion we have the concept of modules graded by G-set: Let A be a C-sets, M=@, M be an Abelian decomposition of the left R-module M. If for VaGA, aEG and VmaEM^. ra GR^. '^'11a EA ,we call M a module graded by G-sets. In chapter 1 section 3 we use the concepts of generalized group action and generalized G-sets generalizing module graded by G-set to module graded byVI7generalized G-sets, and discuss some properties of the category of modules graded by generalized G-sets. The main results are:Lemma 1.3.1Let M.N.PES-gr,consider thecommutative diagram of left R-homomorphisms:feI1oinc^<sup>M.P. If g is S-graded homomorphism( Ii is aS-graded homomorphism), then there exists a S- gradedhomomorphism/i'(g'), sat isF^'ing f^gh'(f= g'h)Theorem 1.3.3 Consider G-graded ring I? and a set of generalized G-set (S⁾,¹,then the category ([J1S,)-gr isequivalent to the category fl,(s,-gr).Corollary 1.3.4 LetR=?^. Rg be a C-graded ring, S he a generalized 6-set, then S-gr is equivalent to the categoryfl (tr(x)-gr) , x runs over a set of representatives of G-orbits of the C-set W.For arbitrary XES, , let er ^mtEc , if s=x,ni. ^g ( g is a formal element), If s^x,rn ^O . VUEZ, letne^, wherey, =ngIfs=xotherwisey, =0. Vr=ⁱ; eR, let re, =(yj,^. :where yc=(^rj)g ifVII...