Algebras, Rings and Modules: Volume 1 by Michiel Hazewinkel

By Michiel Hazewinkel

From the studies of the 1st edition:

"This is the 1st of 2 volumes which goal to take the idea of associative jewelry and their modules from primary definitions to the study frontier. The e-book is written at a degree meant to be obtainable to scholars who've taken normal uncomplicated undergraduate classes in linear algebra and summary algebra. … has been written with significant cognizance to accuracy, and has been proofread with care. … a really welcome function is the vast set of bibliographic and old notes on the finish of every chapter." (Kenneth A. Brown, Mathematical experiences, 2006a)

"This booklet follows within the footsteps of the dear paintings performed through the seventies of systematizing the research of houses and constitution of jewelry by utilizing their different types of modules. … A striking novelty within the current monograph is the learn of semiperfect jewelry via quivers. … one other stable inspiration is the inclusion of the examine of commutative in addition to non-commutative discrete valuation jewelry. every one bankruptcy ends with a few illustrative old notes." (José Gómez Torrecillas, Zentralblatt MATH, Vol. 1086 (12), 2006)

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Additional info for Algebras, Rings and Modules: Volume 1

Example text

Proof. (i) is obvious. , mn ∈ M such that any element m ∈ M has the form m = n mi ai with ai ∈ A. , mn + N generate M/N . 3, we have M/M2 = (M1 ⊕ M2 )/M2 M1 /(M1 ∪ M2 = M1 /0 M1 . Now by (ii) M/M2 can be generated by n elements. Hence, M1 can be generated by n elements. Now we introduce a special class of modules that can be considered as the most natural generalization of vector spaces and that play a very important role in the theory of modules. Definition. , M ⊕ Mi . where Mi AA for all i ∈ I.

Definition. A Boolean algebra is a complemented distributive lattice. It is easy to show that each element of a Boolean algebra has precisely one complement. Indeed, let b and c be complements of an element a. Then b = b ∧ 1 = b ∧ (a ∨ c) = (b ∧ a) ∨ (b ∧ c) = 0 ∨ (b ∧ c) = (a ∧ c) ∨ (b ∧ c) = (a ∨ b) ∧ c = 1 ∧ c = c. We shall use a to denote the complement of an element a in a Boolean algebra. 3. The power set P(S) is a Boolean algebra. 4. Consider the set B = {0, 1} with the ordinary logical operations of disjunction ∨ and conjunction ∧ and operation of complementation 0 = 1 and 1 = 0.

It is a right ideal in A and it is called the right annihilator of the element m. ann(m) = 0, then the element m is called a torsion element, otherwise it is called a torsion-free element. If all elements of an A-module M are torsion, M is called a torsion module. 1 it is easy to obtain the following statement. 2. Every cyclic module is isomorphic to a quotient module of the regular module by some right ideal. Proof. , M = m0 A. We define a map f : A → M by setting f (a) = m0 a. From the module axioms it follows that f is a module homomorphism and, since m0 is the generator of M , we have Im(f ) = M .

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