# Ring Theory: 001 by Author Unknown

By Author Unknown

This is often an abridged version of the author's past two-volume paintings, Ring concept, which concentrates on crucial fabric for a basic ring concept direction whereas ommitting a lot of the cloth meant for ring concept experts. it's been praised by means of reviewers:**"As a textbook for graduate scholars, Ring idea joins the best....The specialists will locate a number of appealing and delightful beneficial properties in Ring conception. the main noteworthy is the inclusion, frequently in supplementations and appendices, of many helpful structures that are difficult to find outdoor of the unique sources....The viewers of nonexperts, mathematicians whose speciality isn't really ring conception, will locate Ring thought best to their needs....They, in addition to scholars, can be good served by way of the various examples of jewelry and the word list of significant results."**--NOTICES OF THE AMS

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This observation is useful in generalizing results about posets. 1’. Define < on Z by a s b iff b divides a. Use this to define a PO on N via exercise 1, and identify this poset with Y(Z). 2. A lattice Y is modular iff Y has the property: If a 5 b and a A c = b A c and a v c = b v c then a = b. (Hint: (-=) Let a , = a v (b A c) and a, = b A (a v c). ) 3. We say b covers a if a < b and there is no x with a < x < b. Using exercise 2 show for any a, b in the modular lattice Y there is a lattice isomorphism from the sublattice {x E 9a < x < a v b to the sublattice {x E 4p:a A b < x < b}, given by x -P x A b, with inverse function y + a v y .

D. =) Reverse the argument Before putting this idea aside, we note an important generalization. Fimod and N < M then N is contained in some maximal submodule M’. Consequently M J M ’ is a simple module in which the image of N is 0. =I:=] Proof;. Write M R x i , where all x i € M . For any chain M , IM , I... of proper submodules, some xi # Mj for all j (since otherwise all xi E Mj for large enough j, implying M = z R x i E Mj contrary to Mi proper). Thus xi 4 M j , which hence must be proper (and is clearly a submodule).

1 Categories of Rings and Modules The language of categories is useful, particularly in certain aspects of module theory. We presuppose a nodding acquaintance with this language; Jacobson [80B, Chapter 13 more than suffices for this purpose. In particular, the reader should know the definition of category, subcategory, (covariant) 14 General Fundamentals functor, contravariant functor, natural transformation (of functors), and natural isomorphism of functors. Given a category W we write Hom,(A, B), or merely Hom(A, B), for the set of morphisms; 1,: A + A denotes the unit morphism.