Algebras, Rings and Modules: Volume 1 by Michiel Hazewinkel

By Michiel Hazewinkel

From the experiences of the 1st edition:

"This is the 1st of 2 volumes which objective to take the idea of associative jewelry and their modules from basic definitions to the study frontier. The booklet is written at a degree meant to be obtainable to scholars who've taken general simple undergraduate classes in linear algebra and summary algebra. … has been written with substantial recognition to accuracy, and has been proofread with care. … a really welcome function is the monstrous set of bibliographic and historic notes on the finish of every chapter." (Kenneth A. Brown, Mathematical reports, 2006a)

"This e-book follows within the footsteps of the dear paintings performed in the course of the seventies of systematizing the research of homes and constitution of jewelry through the use of their different types of modules. … A striking novelty within the current monograph is the learn of semiperfect earrings via quivers. … one other strong suggestion is the inclusion of the learn of commutative in addition to non-commutative discrete valuation earrings. each one bankruptcy ends with a few illustrative historic notes." (José Gómez Torrecillas, Zentralblatt MATH, Vol. 1086 (12), 2006)

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If M and N are A-modules and f : M → N is an A-homomorphism, then M/Ker(f ) Im(f ). PRELIMINARIES 19 Proof. Let m+Ker(f ) be an element of M/Ker(f ). 1, there exists a unique A-homomorphism g : M/Ker(f ) → Imf , where g(m + Ker(f )) = f (m). We need only show that g is an isomorphism. Since every element of Im(f ) has a form f (m) = g(m + Ker(f )), g is an epimorphism. , m ∈ Ker(f ). Therefore m + Ker(f ) = 0 + Ker(f ) is the zero class of M/Ker(f ). Thus, g is a monomorphism. Hence, g is an isomorphism.

But in this case we have the following conditions: ALGEBRAS, RINGS AND MODULES 22 1) πi σi = 1Xi and πi σj = 0 for i = j; 2) if we have a set of elements {mi }, where there is only one element mi ∈ Mi for each i ∈ I, then there exists a unique element m ∈ Mi such that πi m = mi i∈I for each i ∈ I. , An be rings. , n. , bn ) ∈ A. , an bn ). , n. , 1), where the identity of the ring Ai is at the i-th position. , An and denoted by A1 × A2 × ... × An . , 0), where the identity of the ring Ai is at the i-th position and zeroes elsewhere.

Nn where ϕij = πi ϕπj . The elements ϕij are naturally considered as homomorphisms of the module Mj to the module Mi . Let us fix isomorphisms µi : M1 −→ Mi and assign to the matrix ϕ = (ϕij ) the matrix ϕˆ = (µ−1 i ϕij µi ) ∈ Mn (EndA (M1 )). Clearly, this map yields an isomorphism between the rings EndA (M ) and Mn (EndA (M1 )). 2 THE WEDDERBURN-ARTIN THEOREM In this section we shall study a most important class of rings which are called semisimple. Historically the first full classification of rings was obtained for semisimple rings.

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