Later in the book, more advanced topics, such as hereditary rings, categories and functors, flat modules and purity are introduced.
Illinois Journal of Mathematics
These later chapters will also prove a useful reference for researchers in non-commutative ring theory. Enough background material including detailed proofs is supplied to give the student a firm grounding in the subject.
Read more Read less. Review "Dauns No customer reviews. Share your thoughts with other customers. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules generalize ideals and quotients, but he remains unimpressed.
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How do you convince him that studying modules of a ring is a good way to understand that ring? In other words, why does one have to work "external" to the ring? Your answer should also explain why it is a good idea to study a group by studying its representations. In short, I'd tell your friend: "If you believe a ring can be understood geometrically as functions its spectrum, then modules help you by providing more functions with which to measure and characterize its spectrum.
We can talk about the support of a module element, or its vanishing set. More concretely, think of how global sections of a line bundle can act as functions you can use to define map into projective space. When you glue together a module on open sets of a spectrum or a scheme, you get to glue using maps which are module isomorphisms, which are more flexible than the ring isomorphisms required to glue together a scheme.
A fortiori, this certainly doesn't capture the full utility of group representations, but a priori I think it's a decent justification. I want to answer your question twice: first with a "top-down" approach and second with a "bottom-up" approach. Let me limit myself to the first answer here and see how I do. I will take it as mostly self-evident that it is desirable to study groups acting on sets. If you are not doing this -- but thinking of groups only as sets with a certain law of composition -- then you are thinking about groups "in the wrong way".
You are missing out not only on powerful tools for studying abstract groups e. The way to think about a group action is that you have a set S, and it has an automorphism group, Sym S , the group of all bijections from S to itself. More generally, if x is any object in a category C, then it has an automorphism group Aut x , and one can think of a homomorphism from an abstract group G to Aut x as a group action on x. Now in place of a set, we take an abelian group M. This has more structure -- apart from a group Aut M of Z-linear automorphsims, it also has an endomorphism ring End M : the ring of Z-linear maps from M to itself.
Note that End M is in general noncommutative, so this construction is more general than any "ring of functions" construction in commutative! As for rings, this provides a bridge between the abstract notion of a ring and the "real world" notion of endomorphisms of an abelian group.
MA3201 - Rings and Modules
In the first part of the analogy, a distinguished role is played by the category of [left, or right] G-sets for a particular group G. In particular, this provides a way for every group G to be the precise automorphism group of some object in a category: it is the automorphism group of itself. It is of course not true that any abstract group is equal to the full symmetry group Sym S of a set S, so this is an important construction.
In the second part of the analogy, a distinguished role is played by the abelian category of [left, or right] R-modules. I am pretty sure that not every ring is isomorphic to the full endomorphism ring of an abelian group, although this is less obvious than the other case. It might make a good question in its own right This is I think the right "general" answer to the question "Why is studying modules of a ring a good way to understand that ring?
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I'll try that at some future point. It's not clear to me how one would try to prove the first theorem from the definitions of regular. But we also have lots of other structure to work with.
We have all limits and colimits, possibly a tensor product, injective modules which can have a lovely structure theory , duality, etc So not only can we build a lot more objects but one can prove that just the existence of certain objects gives us a lot of information about the ring.
In the commutative with unit case this can be interpreted literally, however, I still think it is relevant suitably adjusted in the case of noncommutative rings. One way to convince the guy would be to make him list interesting questions he can make about rings, and show him which can be solved by looking only at the category of modules.
Somewhere in the middle of that conversation, one should make the point that it may very well happen that two different rings have equivalent categories of modules, so in answering any such question we can change the ring. In homage to Serge Lang, I might suggest that your friend pick up any book on Morita theory and solve all the exercises.
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Less facetiously, I might point out that many interesting ring-theoretic properties can be characterized, sometimes unexpectedly, by properties of the right module category over the ring. This is closely related to Pete Clark's answer, but stated in a slightly different way that I personally find helpful.
I think it's not too hard to convince people that when studying an abstract object, it helps a great deal to be able to "write it down concretely," i. When you write something down concretely, often some structure will emerge that is not immediately evident from the original abstract definition. For example, the character table is an important invariant of a finite group. I don't know how you would come up with this invariant without considering representations of the group. Similarly, modules are representations of rings.
I think the Artin-Wedderburn theorem is a good illustration of the usefulness of considering representations of rings. Even if your interest is only in rings themselves, it's clearly an important result that you can classify all Artinian semisimple rings as products of matrix rings over division rings. If you possess the concept that a ring can be represented as a matrix ring, then it is not too shocking and may even seem natural that something like the Artin-Wedderburn theorem should be true, and moreover you can even see that to prove it you should somehow construct the matrix rings by having the original ring act on something.
Without the concept of a representation or equivalently a module , I don't know how you would proceed; it seems like a difficult and clumsy task at best. So you can disentangle "mixed" notions and work out the concepts more clearly. It's not like embeddings of manifolds were "more interesting" than the theory of manifolds - on the contrary, the gist is distinguishing both.
MA Rings and modules - bocabsterpjozil.gq
The "categorical" approach: Understand an object not via intrinsic properties rather ignore them, they are not part of the categorical data but via the morphisms. By choosing a category to work in, we choose the class of properties that are interesting. Search my Subject Specializations: Select Users without a subscription are not able to see the full content. Cyclic Modules and the Structure of Rings S. Jain, Ashish K. Srivastava, and Askar A. Tuganbaev Abstract This book provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property.
More This book provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. Authors Affiliations are at time of print publication. Print Save Cite Email Share.
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