By Michiel Hazewinkel
From the experiences of the 1st edition:
"This is the 1st of 2 volumes which target to take the idea of associative jewelry and their modules from basic definitions to the learn frontier. The booklet is written at a degree meant to be obtainable to scholars who've taken ordinary easy 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 publication follows within the footsteps of the dear paintings performed through the seventies of systematizing the research of homes and constitution of earrings through the use of their different types of modules. … A impressive novelty within the current monograph is the examine of semiperfect jewelry via quivers. … one other strong notion is the inclusion of the research of commutative in addition to non-commutative discrete valuation jewelry. each one bankruptcy ends with a few illustrative historic notes." (José Gómez Torrecillas, Zentralblatt MATH, Vol. 1086 (12), 2006)
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Additional info for Algebras, Rings and Modules: Volume 1
Since Ker(f ) = 0, we have x = 0. Therefore M1 ∩ M2 = 0. 2) ⇒ 1). Conversely, let M = M1 + M2 and M1 ∩ M2 = 0, then obviously f is an epimorphism. , x = −y. , Ker(f ) = 0. , f is an isomorphism. Inspired by this proposition we may introduce the following deﬁnition. 1 are satisﬁed. The submodules M1 and M2 are called direct summands of the module M . The internal direct sum of several modules can be deﬁned in a similar way. For this purpose we shall prove the following statement. 2. Let Mi (i ∈ I) be a family of submodules of a module M , and f : ⊕ Mi → M be the homomorphism deﬁned by the formula f (⊕i mi ) = i mi .
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. , a vector space. Free modules play an important role in the theory of modules.
1. , mn of M such that every element m ∈ M can be written n as m = mi ai , where ai ∈ A. i=1 The following lemma gives some simple but useful properties of ﬁnitely generated modules. 1. If M is an A-module then: (i) If M is a sum of the ﬁnite number of ﬁnitely generated modules, then M is a ﬁnitely generated module. (ii) If M can be generated by n elements and N is a submodule of M , then M/N can be generated by n elements. PRELIMINARIES 25 (iii) If M = M1 ⊕ M2 and M can be generated by n elements, then M1 can be generated by n elements.
Algebras, Rings and Modules: Volume 1 by Michiel Hazewinkel