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October 29th, 2013, 03:46 PM  #1 
Newbie Joined: Nov 2011 Posts: 9 Thanks: 0  Abelian XGroups and Noetherian (Abelian) XGroups
I was having a quick look at Isaacs : Algebra  A Graduate Course and was interested in his approach to Noetherian modules. I wonder though how standard is his treatment and his terminology. Is this an accepted way to study module theory and is his term XGroup fairly standard (glimpsing at other books it does not seem to be!) and, further, if the structure he is talking about is a standard item of study, is his terminology "XGroup" standard? If not, what is the usual terminology. A bit of information on Isaacs treatment of XGroups follows: In Chapter 10: Operator Groups and Unique Decompositions, on page 129 (see attachment) Isaacs defines an XGroup as follows: 0.1 DEFINITION. Let X be an arbitrary (possibly empty) set and Let G be a group. We say that G is an Xgroup (or group with operator set X) provided that for each and , there is defined an element such that if then I am not quite sure what the "operator set" is, but from what I can determine the notation refers to the conjugate of g with respect to x (this is defined on page 20  see attachment) In Chapter 10: Module Theory without Rings, Isaacs defines abelian Xgroups and uses them to develop module theory and in particular Noetherian and Artinian Xgroups. Regarding a Noetherian (abelian) Xgroup, the definition (Isaacs page 146) is as follows: DEFINITION. Let M be an abelian Xgroup and consider the poset of all Xgroups ordered by the inclusion $\displaystyle \supseteq $. We say M is Noetherian if this poset satisfies the ACC (ascending chain condition) My question is  is this a standard and accepted way to introduce module theory and the theory of Noetherian and Artinian modules and rings. Further, can someone give a couple of simple and explicit examples of Xgroups in which the sets X and G are spelled out and some example operations are shown. Peter 

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abelian, noetherian, xgroups 
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