Download PDF by Michiel Hazewinkel, Nadiya M. Gubareni: Algebras, Rings and Modules: Non-commutative Algebras and

By Michiel Hazewinkel, Nadiya M. Gubareni

The idea of algebras, earrings, and modules is likely one of the primary domain names of contemporary arithmetic. common algebra, extra particularly non-commutative algebra, is poised for significant advances within the twenty-first century (together with and in interplay with combinatorics), simply as topology, research, and chance skilled within the 20th century. This quantity is a continuation and an in-depth learn, stressing the non-commutative nature of the 1st volumes of Algebras, earrings and Modules via M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. it's mostly autonomous of the opposite volumes. The proper structures and effects from prior volumes were offered during this quantity.

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A ring is right (left) serial if it is a right (left) serial module over itself. A ring which is both a right and left serial ring is said to be a serial ring. Recall that a module M is finitely presented if M is a quotient of a finitely generated free module with finitely generated kernel. The following well known theorem characterizes serial rings in terms of finitely presented modules. 1. (Drozd-Warfield Theorem, [67], [320]). g. ) For a ring A the following conditions are equivalent: 1. A is serial; 2.

Proof. 1 there exists a unique mapping ϕ : B → A such that πi ϕ = gi . So it only remains to show that ϕ is a ring morphism which follows from the fact that gi and πi are ring morphisms for each i. Indeed, for any b1 , b2 ∈ B one has πi ϕ(b1 +b2 ) = gi (b1 +b2 ) = gi (b1 )+gi (b2 ) = πi ϕ(b1 )+πi ϕ(b2 ) = πi (ϕ(b1 )+ϕ(b2 )) whence ϕ(b1 + b2 ) = ϕ(b1 ) + ϕ(b2 ). , ϕ is a ring morphism. 4. The direct product is sometimes called the complete direct product to distinguish it from the discrete direct product (or direct sum).

In i=1 X i . Point i=1 out that this is not the case in a category of rings. The direct product of a finite number of rings were considered in [146]. Now consider this construction for an infinite number of rings. 2. Let { Ai }i ∈I , be a family of rings. The direct product of the rings { Ai }i ∈I is the Cartesian product A = Ai with operations of addition and i ∈I multiplication defined componentwise: ( f + g)(i) = f (i) + g(i) ( f g)(i) = f (i)g(i) for any f , g ∈ A and for each i ∈ I. 1Recall that a category C is Abelian if its morphisms and objects can be added, there exist kernels and cokernels, and moreover, every monomorphism is a kernel of some morphism and every epimorphism is a cokernel of some morphism.

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