By A. J. Berrick
This concise advent to ring thought, module conception and quantity idea is perfect for a primary yr graduate scholar, in addition to being a superb reference for operating mathematicians in different parts. ranging from definitions, the ebook introduces basic structures of earrings and modules, as direct sums or items, and via certain sequences. It then explores the constitution of modules over numerous kinds of ring: noncommutative polynomial jewelry, Artinian jewelry (both semisimple and not), and Dedekind domain names. It additionally indicates how Dedekind domain names come up in quantity concept, and explicitly calculates a few jewelry of integers and their category teams. approximately 2 hundred workouts supplement the textual content and introduce extra themes. This publication offers the historical past fabric for the authors' imminent significant other quantity different types and Modules. Armed with those texts, the reader can be prepared for extra complex themes in K-theory, homological algebra and algebraic quantity idea.
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Extra info for An Introduction to Rings and Modules With K-theory in View
Example text
2), an abelian group is essentially the same thing as a Z-module. Here, we give some first results on the -decomposability of abelian groups which will be developed in the exercises-to-this section, and in much greater generality in our discussion of Dedekind domains in Chapter 6. -module. A (right) submodule of Z is the same as an ideal, and- so has the -form aZ for some integer a. Since aZ n bZ 0 for any two nonzero ideals, it follows that Z. is- indecomposable. (ii) Let p be a prime number and r be a natural number.
It is easily verified that the sum is also a submodule of M. This definition can be extended to an infinite set {Mi Ij E I} of submodules: E M = {E mi I mi E and mi = 0 for almost all i}. iEI Again, it is straightforward to show that the _sum is a -submodule of M. Note that the set-theoretic intersection RE/ M is_always a submodule of M, regardless of whether or not the index-set I is finite. It is often convenient to specify _a module or one aits submodules irr terms of generators. Given an element x of a module M, x1:1-=-{xr I r E R} is the cyclic subrnodule generated by-x, -and M itself is -cyclic if M = xR for some x.
Thus, if we take S to be the set of proper submodules of a right module M, a maximal element of S is a maximal submodule of M. 3) can be interpreted in the same way. 5). Zorn's Lemma gives a convenient sufficient condition for their existence. We need a preliminary definition. X--< p -and p. < v, then A < v. - We allow the empty set 0 to be regarded as an ordered set and we may write A > in place of p < A. totally ordered' is used when we wish to emphasize the difference between ordered sets and partially ordered sets, which need not satisfy (T01).